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<title>Inequalities of Jensen Type for \(AH\)-Convex Functions\(^\ast \): Inequalities of Jensen Type for \(AH\)-Convex Functions\(^\ast \)</title>
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<h1>Inequalities of Jensen Type for \(AH\)-Convex Functions\(^\ast \)</h1>
<p class="authors">
<span class="author">Silvestru Sever Dragomir\(^\S \)</span>
</p>
<p class="date">May 23rd, 2016.</p>
</div>
<p>\(^\S \)Department of Mathematics, College of Engineering &amp; Science, Victoria University, Melbourne City, Victoria, Australia, e-mail: <span class="tt">sever.dragomir@vu.edu.au</span>. </p>

<div class="abstract"><p> Some integral inequalities of Jensen type for \(AH\)-convex functions defined on intervals of real line are given. Applications for power and logarithm functions are provided as well. Some inequalities for functions of selfadjoint operators in Hilbert spaces are also established. </p>
<p><b class="bf">MSC.</b> 26D15; 25D10, 47A63. </p>
<p><b class="bf">Keywords.</b> Convex functions, Integral inequalities, \(AH\)-Convex functions. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\left( \Omega ,\mathcal{A},\mu \right) \) be a measurable space consisting of a set \(\Omega ,\) a \(\sigma \)-algebra \(\mathcal{A}\) of parts of \(\Omega \) and a countably additive and positive measure \(\mu \) on \(\mathcal{A}\) with values in \(\mathbb {R}\cup \left\{  \infty \right\}  .\) For a \(\mu \)-measurable function \(w:\Omega \rightarrow \mathbb {R}\), with \(w\left( x\right) \geq 0\) for \(\mu \)-a.e. (almost every) \(x\in \Omega ,\) consider the Lebesgue space </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  L_{w}\left( \Omega ,\mu \right) :=\Big\{ f:\Omega \rightarrow \mathbb {R},\; f\text{ is }\mu \text{-measurable and }\int _{\Omega }w\left( x\right) \left\vert f\left( x\right) \right\vert d\mu \left( x\right) {\lt}\infty \Big\} . \end{equation*}
</div>
<p>For simplicity of notation we write everywhere in the sequel \(\int _{\Omega }wd\mu \) instead of \(\int _{\Omega }w\left( x\right) d\mu \left( x\right) .\) </p>
<p>If \(f,\) \(g:\Omega \rightarrow \mathbb {R}\) are \(\mu \)-measurable functions and \(f,\) \(g,\) \(fg\in L_{w}\left( \Omega ,\mu \right) ,\) then we may consider the <i class="itshape">Čebyšev functional</i> </p>
<div class="equation" id="a.1">
<p>
  <div class="equation_content">
    \begin{equation}  T_{w}\left( f,g\right) :=\int _{\Omega }wfgd\mu -\int _{\Omega }wfd\mu \int _{\Omega }wgd\mu . \label{a.1} \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p>The following result is known in the literature as the <i class="itshape">Grüss inequality</i> </p>
<div class="equation" id="a.2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\vert T_{w}\left( f,g\right) \right\vert \leq \tfrac {1}{4}\left( \Gamma -\gamma \right) \left( \Delta -\delta \right) , \label{a.2} \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p>provided </p>
<div class="equation" id="a.3">
<p>
  <div class="equation_content">
    \begin{equation}  -\infty <\gamma \leq f\left( x\right) \leq \Gamma <\infty ,\; \; -\infty <\delta \leq g\left( x\right) \leq \Delta <\infty \label{a.3} \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p>for \(\mu \)-a.e. (almost every) \(x\in \Omega .\) </p>
<p>The constant \(\frac{1}{4}\) is sharp in the sense that it cannot be replaced by a smaller constant. </p>
<p>If we assume that \(-\infty {\lt}\gamma \leq f\left( x\right) \leq \Gamma {\lt}\infty \) for \(\mu \)-a.e. \(x\in \Omega ,\) then by the Grüss inequality for \(g=f\) and by the Schwarz’s integral inequality, we have </p>
<div class="displaymath" id="b.18">
  \begin{align} &  \int _{\Omega }w\left\vert f-\int _{\Omega }wfd\mu \right\vert d\mu \leq \label{b.18} \\ &  \leq \left[ \int _{\Omega }wf^{2}d\mu -\left( \int _{\Omega }wfd\mu \right) ^{2}\right] ^{\frac{1}{2}}\leq \tfrac {1}{2}\left( \Gamma -\gamma \right) . \notag \end{align}
</div>
<p>In order to provide a reverse of the celebrated Jensen’s integral inequality for convex functions, S.S. Dragomir obtained in 2002 <span class="cite">
	[
	<a href="#D1" >6</a>
	]
</span> the following result: </p>
<p><div class="theorem_thmwrapper " id="t.A">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\Phi :\left[ m,M\right] \subset \mathbb {R\rightarrow R}\) be a differentiable convex function on \(\left( m,M\right) \) and \(f:\Omega \rightarrow \left[ m,M\right] \) so that \(\Phi \circ f,\) \(f,\) \(\Phi ^{\prime }\circ f,\; \left( \Phi ^{\prime }\circ f\right) f\in L_{w}\left( \Omega ,\mu \right) ,\) where \(w\geq 0\) \(\mu \)-a.e. (almost everywhere) on \(\Omega \) with \(\int _{\Omega }wd\mu =1.\) Then we have the inequality: </p>
<div class="displaymath" id="e.1">
  \begin{align}  0&  \leq \int _{\Omega }\left( \Phi \circ f\right) wd\mu -\Phi \left( \int _{\Omega }fwd\mu \right) \label{e.1} \\ &  \leq \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu -\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu \int _{\Omega }wfd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \int _{\Omega }w\left\vert f-\int _{\Omega }fwd\mu \right\vert d\mu . \notag \end{align}
</div>

  </div>
</div> </p>
<p>For a generalization of the first inequality in (<a href="#e.1">1.5</a>) without the differentiability assumption and the derivative \(\Phi ^{\prime }\) replaced with a selection \(\varphi \) from the subdifferential \(\partial \Phi \), see the paper <span class="cite">
	[
	<a href="#N" >18</a>
	]
</span> by C. P. Niculescu. </p>
<p>If \(\mu \left( \Omega \right) {\lt}\infty \) and \(\Phi \circ f,\) \(f,\) \(\Phi ^{\prime }\circ f,\; \left( \Phi ^{\prime }\circ f\right) \cdot f\in L\left( \Omega ,\mu \right) ,\) then we have the inequality: </p>
<div class="displaymath" id="e.2">
  \begin{align}  0&  \leq \tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }\left( \Phi \circ f\right) d\mu -\Phi \left( \tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }fd\mu \right) \label{e.2} \\ &  \leq \tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fd\mu -\tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) d\mu \cdot \tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }fd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }\left\vert f-\tfrac {1}{\mu \left( \Omega \right) }\int _{\Omega }fd\mu \right\vert d\mu . \notag \end{align}
</div>
<p>The following discrete inequality is of interest as well. </p>
<p><div class="corollary_thmwrapper " id="c5.5">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">1.2</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(\Phi :\left[ m,M\right] \rightarrow \mathbb {R}\) be a differentiable convex function on \(\left( m,M\right) .\) If \(x_{i}\in \left[ m,M\right] \) and \(w_{i}\geq 0\) \(\left( i=1,\dots ,n\right) \) with \(W_{n}:=\sum _{i=1}^{n}w_{i}=1,\) then one has the counterpart of Jensen’s weighted discrete inequality: </p>
<div class="displaymath" id="5.6">
  \begin{align}  0&  \leq \sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) -\Phi \left( \sum _{i=1}^{n}w_{i}x_{i}\right) \label{5.6} \\ &  \leq \sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) x_{i}-\sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) \sum _{i=1}^{n}w_{i}x_{i} \notag \\ &  \leq \tfrac {1}{2}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \sum _{i=1}^{n}w_{i}\left\vert x_{i}-\sum _{j=1}^{n}w_{j}x_{j}\right\vert . \notag \end{align}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000004">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>We notice that the inequality between the first and the second term in (<a href="#5.6">1.7</a>) was proved in 1994 by Dragomir &amp; Ionescu, see <span class="cite">
	[
	<a href="#DI" >12</a>
	]
</span>. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>On making use of the results (<a href="#e.1">1.5</a>) and (<a href="#b.18">1.4</a>), we can state the following sequence of reverse inequalities</p>
<div class="displaymath" id="b.19">
  \begin{align}  0&  \leq \int _{\Omega }\left( \Phi \circ f\right) wd\mu -\Phi \left( \int _{\Omega }fwd\mu \right) \label{b.19} \\ &  \leq \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu -\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu \int _{\Omega }fwd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \int _{\Omega }\left\vert f-\int _{\Omega }wfd\mu \right\vert wd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \left[ \int _{\Omega }f^{2}wd\mu -\left( \int _{\Omega }fwd\mu \right) ^{2}\right] ^{\frac{1}{2}} \notag \\ &  \leq \tfrac {1}{4}\left[ \Phi ^{\prime }\left( M\right) -\Phi ^{\prime }\left( m\right) \right] \left( M-m\right) , \notag \end{align}
</div>
<p>provided that \(\Phi :\left[ m,M\right] \subset \mathbb {R\rightarrow R}\) is a differentiable convex function on \(\left( m,M\right) \) and \(f:\Omega \rightarrow \left[ m,M\right] \) so that \(\Phi \circ f,\) \(f,\) \(\Phi ^{\prime }\circ f,\; \left( \Phi ^{\prime }\circ f\right) f\in L_{w}\left( \Omega ,\mu \right) ,\) where \(w\geq 0\) \(\mu \)-a.e. on \(\Omega \) with \(\int _{\Omega }wd\mu =1.\) </p>
<p><div class="remark_thmwrapper " id="a0000000005">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.4</span>
  </div>
  <div class="remark_thmcontent">
  <p>We notice that the inequality between the first, second and last term from (<a href="#b.19">1.8</a>) was proved in the general case of positive linear functionals in 2001 by S.S. Dragomir in <span class="cite">
	[
	<a href="#D" >5</a>
	]
</span>. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>The following reverse of the Jensen’s inequality holds <span class="cite">
	[
	<a href="#SSD1" >9</a>
	]
</span>, <span class="cite">
	[
	<a href="#SSD2" >10</a>
	]
</span>: </p>
<p><div class="theorem_thmwrapper " id="t.2.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\Phi :I\rightarrow \mathbb {R}\) be a continuous convex function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R}\), \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), \(\mathring {I}\) is the interior of \(I.\) If \(f:\Omega \rightarrow \mathbb {R}\) is \(\mu \)-measurable, satisfies the bounds</p>
<div class="equation" id="Con">
<p>
  <div class="equation_content">
    \begin{equation}  -\infty <m\leq f\left( x\right) \leq M<\infty \text{ \hspace{0.05in}for \hspace{0.05in}}\mu \text{-a.e. }x\in \Omega \label{Con} \end{equation}
  </div>
  <span class="equation_label">1.9</span>
</p>
</div>
<p>and such that \(f,\Phi \circ f\in L_{w}\left( \Omega ,\mu \right) ,\) where \(w\geq 0\) \(\mu \)-a.e. on \(\Omega \) with \(\int _{\Omega }wd\mu =1\), then</p>
<div class="displaymath" id="e.2.1">
  \begin{align}  0&  \leq \int _{\Omega }\left( \Phi \circ f\right) wd\mu -\Phi \left( \bar{f}_{\Omega ,w}\right) \label{e.2.1} \\ &  \leq \tfrac {( M-\bar{f}_{\Omega ,w}) ( \bar{f}_{\Omega ,w}-m) }{M-m}\sup _{t\in \left( m,M\right) }\Psi _{\Phi }\left( t;m,M\right) \notag \\ &  \leq \left( M-\bar{f}_{\Omega ,w}\right) \left( \bar{f}_{\Omega ,w}-m\right) \tfrac {\Phi _{-}^{\prime }\left( M\right) -\Phi _{+}^{\prime }\left( m\right) }{M-m} \notag \\ &  \leq \tfrac {1}{4}\left( M-m\right) \left[ \Phi _{-}^{\prime }\left( M\right) -\Phi _{+}^{\prime }\left( m\right) \right] , \notag \end{align}
</div>
<p>where \(\bar{f}_{\Omega ,w}:=\int _{\Omega }w\left( x\right) f\left( x\right) d\mu \left( x\right) \in \left[ m,M\right] \) and \(\Psi _{\Phi }\left( \cdot ;m,M\right) :\left( m,M\right) \rightarrow \mathbb {R}\) is defined by</p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  \Psi _{\Phi }\left( t;m,M\right) =\frac{\Phi \left( M\right) -\Phi \left( t\right) }{M-t}-\frac{\Phi \left( t\right) -\Phi \left( m\right) }{t-m}. \end{equation*}
</div>
<p>We also have the inequality</p>
<div class="displaymath" id="e.2.1.a">
  \begin{align}  0&  \leq \int _{\Omega }\left( \Phi \circ f\right) wd\mu -\Phi \left( \bar{f}_{\Omega ,w}\right) \leq \tfrac {1}{4}\left( M-m\right) \Psi _{\Phi }\left( \bar{f}_{\Omega ,w};m,M\right) \label{e.2.1.a} \\ &  \leq \tfrac {1}{4}\left( M-m\right) \left[ \Phi _{-}^{\prime }\left( M\right) -\Phi _{+}^{\prime }\left( m\right) \right] , \notag \end{align}
</div>
<p>provided that \(\bar{f}_{\Omega ,w}\in \left( m,M\right) .\) </p>

  </div>
</div> </p>
<p>For a real function \(g:\left[ m,M\right] \rightarrow \mathbb {R}\) and two distinct points \(\alpha ,\beta \in \left[ m,M\right] \) we recall that the <i class="itshape">divided difference</i> of \(g\) in these points is defined by</p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  \left[ \alpha ,\beta ;g\right] :=\frac{g\left( \beta \right) -g\left( \alpha \right) }{\beta -\alpha }. \end{equation*}
</div>
<p>In what follows, we assume that \(w:\Omega \rightarrow \mathbb {R}\), with \(w\left( x\right) \geq 0\) for \(\mu \)-a.e. \(x\in \Omega ,\) is a \(\mu \)-measurable function with \(\int _{\Omega }wd\mu =1.\) </p>
<p>The following result holds <span class="cite">
	[
	<a href="#SSD3" >11</a>
	]
</span>: </p>
<p><div class="theorem_thmwrapper " id="t.2.1.a">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\Phi :I\rightarrow \mathbb {R}\) be a continuous convex function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R},\) \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), \(\mathring {I}\) the interior of \(I.\) If \(f:\Omega \rightarrow \mathbb {R},\; \)is \(\mu \)-measurable, satisfying the bounds (<a href="#Con">1.9</a>) and such that \(f,\) \(\Phi \circ f\in L_{w}\left( \Omega ,\mu \right) ,\) then by denoting</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  \overline{f}_{\Omega ,w}:=\int _{\Omega }wfd\mu \in \left[ m,M\right] \end{equation*}
</div>
<p>and assuming that \(\overline{f}_{\Omega ,w}\neq m,M,\) we have</p>
<div class="displaymath" id="e.2.0.1">
  \begin{align} &  \left\vert \int _{\Omega }\left\vert \Phi \left( f\right) -\Phi \left( \overline{f}_{\Omega ,w}\right) \right\vert \operatorname {sgn}\left[ f-\overline{f}_{\Omega ,w}\right] wd\mu \right\vert \leq \label{e.2.0.1} \\ &  \leq \int _{\Omega }\left( \Phi \circ f\right) wd\mu -\Phi \left( \overline{f}_{\Omega ,w}\right) \notag \\ &  \leq \tfrac {1}{2}\left( \left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) D_{w}\left( f\right) \notag \\ &  \leq \tfrac {1}{2}\left( \left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) D_{w,2}\left( f\right) \notag \\ &  \leq \tfrac {1}{4}\left( \left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) \left( M-m\right) , \notag \end{align}
</div>
<p>where</p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  D_{w}\left( f\right) :=\int _{\Omega }w\left\vert f-\overline{f}_{\Omega ,w}\right\vert d\mu \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  D_{w,2}\left( f\right) :=\left[ \int _{\Omega }wf^{2}d\mu -\left( \overline{f}_{\Omega ,w}\right) ^{2}\right] ^{\frac{1}{2}} \end{equation*}
</div>
<p>The constant \(\frac{1}{2}\) in the second inequality from (<a href="#e.2.1">1.10</a>) is the best possible. </p>

  </div>
</div> </p>
<p>For recent results related to Jensen’s inequality, see <span class="cite">
	[
	<a href="#A" >1</a>
	]
</span>–<span class="cite">
	[
	<a href="#SSDBul" >8</a>
	]
</span>, <span class="cite">
	[
	<a href="#H" >13</a>
	]
</span>–<span class="cite">
	[
	<a href="#ZK" >23</a>
	]
</span> and the references therein. </p>
<p>Motivated by the above results, in this paper we establish some Jensen type inequalities for the class of \(AH\)-<i class="itshape">convex (concave) </i>functions. Some applications for power and logarithmic functions are provided as well. Some inequalities for functions of selfadjoint operators in Hilbert spaces are also established. </p>
<h1 id="a0000000011">2 \(AH\)-Convex Functions</h1>
<p>Let \(X\) be a linear space and \(C\) a convex subset in \(X.\) A function \(\Phi :C\rightarrow \mathbb {R\setminus }\left\{  0\right\}  \) is called \(AH\)-<i class="itshape">convex (concave)</i> on the convex set \(C\) if the following inequality holds</p>
<div class="equation" id="AH">
<p>
  <div class="equation_content">
    \begin{equation}  \Phi \left( \left( 1-\lambda \right) x+\lambda y\right) \leq \left( \geq \right) \frac{1}{\left( 1-\lambda \right) \frac{1}{\Phi \left( x\right) }+\lambda \frac{1}{\Phi \left( y\right) }}=\frac{\Phi \left( x\right) \Phi \left( y\right) }{\left( 1-\lambda \right) \Phi \left( y\right) +\lambda \Phi \left( x\right) } \tag {AH} \label{AH} \end{equation}
  </div>
  <span class="equation_label">2.13</span>
</p>
</div>
<p>for any \(x,y\in C\) and \(\lambda \in \left[ 0,1\right] .\) </p>
<p>An important case which provides many examples is that one in which the function is assumed to be positive for any \(x\in C.\) In that situation the inequality (<a href="#AH">2.13</a>) is equivalent to</p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  \left( 1-\lambda \right) \frac{1}{\Phi \left( x\right) }+\lambda \frac{1}{\Phi \left( y\right) }\leq \left( \geq \right) \frac{1}{\Phi \left( \left( 1-\lambda \right) x+\lambda y\right) } \end{equation*}
</div>
<p>for any \(x,y\in C\) and \(\lambda \in \left[ 0,1\right] .\) </p>
<p>Therefore we can state the following fact: </p>
<p><div class="criterion_thmwrapper " id="Cr.1">
  <div class="criterion_thmheading">
    <span class="criterion_thmcaption">
    Criterion
    </span>
    <span class="criterion_thmlabel">2.1</span>
  </div>
  <div class="criterion_thmcontent">
  <p>Let \(X\) be a linear space and \(C\) a convex subset in \(X.\) The function \(\Phi :C\rightarrow \left( 0,\infty \right) \) is \(AH\)-convex (concave) on \(C\) if and only if \(\frac{1}{\Phi }\) is concave (convex) on \(C\) in the usual sense. </p>

  </div>
</div> </p>
<p>In what follows, we assume that \(w:\Omega \rightarrow \mathbb {R}\), with \(w\left( x\right) \geq 0\) for \(\mu \)-a.e. \(x\in \Omega ,\) is a \(\mu \)-measurable function with \(\int _{\Omega }wd\mu =1.\) </p>
<p>If \(\Delta :I\rightarrow \mathbb {R}\) is continuous concave function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R},\) \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), \(\mathring {I}\) the interior of \(I,\) then by taking \(\Phi =-\Delta \) in (<a href="#e.2.0.1">1.12</a>) we get </p>
<div class="displaymath" id="e.3.0">
  \begin{align} &  \left\vert \int _{\Omega }\left\vert \Delta \left( f\right) -\Delta \left( \overline{f}_{\Omega ,w}\right) \right\vert \operatorname {sgn}\left[ f-\overline{f}_{\Omega ,w}\right] wd\mu \right\vert \leq \label{e.3.0} \\ &  \leq \Delta \left( \overline{f}_{\Omega ,w}\right) -\int _{\Omega }\left( \Delta \circ f\right) wd\mu \notag \\ &  \leq \tfrac {1}{2}\left( \left[ m,\overline{f}_{\Omega ,w};\Delta \right] -\left[ \overline{f}_{\Omega ,w},M;\Delta \right] \right) D_{w}\left( f\right) \notag \\ &  \leq \tfrac {1}{2}\left( \left[ m,\overline{f}_{\Omega ,w};\Delta \right] -\left[ \overline{f}_{\Omega ,w},M;\Delta \right] \right) D_{w,2}\left( f\right) \notag \\ &  \leq \tfrac {1}{4}\left( \left[ m,\overline{f}_{\Omega ,w};\Delta \right] -\left[ \overline{f}_{\Omega ,w},M;\Delta \right] \right) \left( M-m\right) . \notag \end{align}
</div>
<p>If \(\Phi :I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R},\) \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), then by taking \(\Delta =\frac{1}{\Phi }\) in (<a href="#e.3.0">2.14</a>) we get </p>
<div class="displaymath" id="a0000000013">
  \begin{align*} &  \left\vert \int _{\Omega }\left\vert \tfrac {\Phi ( \overline{f}_{\Omega ,w}) }{\Phi \left( f\right) }-1\right\vert \operatorname {sgn}\left[ f-\overline{f}_{\Omega ,w}\right] wd\mu \right\vert \leq \\ &  \leq 1-\Phi \left( \overline{f}_{\Omega ,w}\right) \int _{\Omega }\tfrac {wd\mu }{\Phi \circ f} \\ &  \leq \tfrac {1}{2}\left( \tfrac {1}{\Phi \left( M\right) }\left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\tfrac {1}{\Phi \left( m\right) }\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) D_{w}\left( f\right) \\ &  \leq \tfrac {1}{2}\left( \tfrac {1}{\Phi \left( M\right) }\left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\tfrac {1}{\Phi \left( m\right) }\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) D_{w,2}\left( f\right) \\ &  \leq \tfrac {1}{4}\left( \tfrac {1}{\Phi \left( M\right) }\left[ \overline{f}_{\Omega ,w},M;\Phi \right] -\tfrac {1}{\Phi \left( m\right) }\left[ m,\overline{f}_{\Omega ,w};\Phi \right] \right) \left( M-m\right) , \end{align*}
</div>
<p>provided that \(f:\Omega \rightarrow \mathbb {R},\; \)is \(\mu \)-measurable, satisfying the bounds (<a href="#Con">1.9</a>) and such that \(f,\left( \Phi \circ f\right) ^{-1}\in L_{w}\left( \Omega ,\mu \right) .\) As above</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  \overline{f}_{\Omega ,w}:=\int _{\Omega }wfd\mu \in \left[ m,M\right] \end{equation*}
</div>
<p>and we assume that \(\overline{f}_{\Omega ,w}\neq m,M,\) </p>
<p>In the case of functions defined on real line we have: </p>
<p><div class="proposition_thmwrapper " id="p.3.1">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2.2</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(\Phi :I\rightarrow \left( 0,\infty \right) \) be defined on the interval \(I.\) The following statements are equivalent: </p>
<p>(i) The function \(\Phi \) is \(AH\)-convex (concave) on \(I;\) </p>
<p>(ii) For any \(x,y\in \mathring {I}\), the interior of \(I,\) then there exists \(\varphi \left( y\right) \in [ \Phi _{-}^{\prime }\left( y\right) ,\) \(\Phi _{+}^{\prime }\left( y\right) ] \) such that </p>
<div class="equation" id="e.3.3.a">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\Phi \left( y\right) }{\Phi \left( x\right) }-1\leq \left( \geq \right) \frac{\varphi \left( y\right) }{\Phi \left( y\right) }\left( y-x\right) \label{e.3.3.a} \end{equation}
  </div>
  <span class="equation_label">2.15</span>
</p>
</div>
<p>holds. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> (i) \(\Rightarrow \) (ii). Let \(x,y\in \mathring {I}.\) Since the function \(\frac{1}{\Phi }\) is concave (convex) then the lateral derivatives \(\Phi _{-}^{\prime }\left( y\right) ,\Phi _{+}^{\prime }\left( y\right) \) exists for \(y\in \mathring {I}\) and \(\left( \frac{1}{\Phi }\right) _{-\left( +\right) }^{\prime }\left( y\right) =-\frac{\Phi _{-\left( +\right) }^{\prime }\left( y\right) }{\Phi ^{2}\left( y\right) }.\) </p>
<p>Since \(\frac{1}{\Phi }\) is concave (convex) then we have the gradient inequality</p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  \frac{1}{\Phi \left( y\right) }-\frac{1}{\Phi \left( x\right) }\geq \left( \leq \right) \lambda \left( y\right) \left( y-x\right) =-\lambda \left( y\right) \left( x-y\right) \end{equation*}
</div>
<p>with \(\lambda \left( y\right) \in \left[ -\frac{\Phi _{+}^{\prime }\left( y\right) }{\Phi ^{2}\left( y\right) },-\frac{\Phi _{-}^{\prime }\left( y\right) }{\Phi ^{2}\left( y\right) }\right] ,\) which is equivalent to</p>
<div class="equation" id="e.3.4">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{1}{\Phi \left( y\right) }-\frac{1}{\Phi \left( x\right) }\geq \left( \leq \right) \frac{\varphi \left( y\right) }{\Phi ^{2}\left( y\right) }\left( x-y\right) \label{e.3.4} \end{equation}
  </div>
  <span class="equation_label">2.16</span>
</p>
</div>
<p>with \(\varphi \left( y\right) \in \left[ \Phi _{-}^{\prime }\left( y\right) ,\Phi _{+}^{\prime }\left( y\right) \right] .\) </p>
<p>The inequality (<a href="#e.3.4">2.16</a>) can be also written as </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  1-\frac{\Phi \left( y\right) }{\Phi \left( x\right) }\geq \left( \leq \right) \frac{\varphi \left( y\right) }{\Phi \left( y\right) }\left( x-y\right) \end{equation*}
</div>
<p>or as</p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  \frac{\Phi \left( y\right) }{\Phi \left( x\right) }-1\leq \left( \geq \right) \frac{\varphi \left( y\right) }{\Phi \left( y\right) }\left( y-x\right) \end{equation*}
</div>
<p>and the inequality (<a href="#e.3.3.a">2.15</a>) is proved. </p>
<p>“(ii) \(\Rightarrow \) (i)" Let \(x,y\in I\) and \(\lambda \in \left( 0,1\right) . \) Then \(\left( 1-\lambda \right) x+\lambda y\in \mathring {I}.\) From (<a href="#e.3.4">2.16</a>) we have</p>
<div class="displaymath" id="e.3.4.a">
  \begin{align}  \frac{1}{\Phi \left( x\right) }-\frac{1}{\Phi \left( \left( 1-\lambda \right) x+\lambda y\right) }&  \geq \left( \leq \right) \frac{\varphi \left( \left( 1-\lambda \right) x+\lambda y\right) }{\Phi ^{2}\left( \left( 1-\lambda \right) x+\lambda y\right) }\left( \left( 1-\lambda \right) x+\lambda y-x\right) \label{e.3.4.a} \\ &  =\lambda \left( y-x\right) \frac{\varphi \left( \left( 1-\lambda \right) x+\lambda y\right) }{\Phi ^{2}\left( \left( 1-\lambda \right) x+\lambda y\right) } \notag \end{align}
</div>
<p>and</p>
<div class="displaymath" id="e.3.4.b">
  \begin{align}  \frac{1}{\Phi \left( y\right) }-\frac{1}{\Phi \left( \left( 1-\lambda \right) x+\lambda y\right) }&  \geq \left( \leq \right) \frac{\varphi \left( \left( 1-\lambda \right) x+\lambda y\right) }{\Phi \left( \left( 1-\lambda \right) x+\lambda y\right) }\left( \left( 1-\lambda \right) x+\lambda y-y\right) \label{e.3.4.b} \\ &  =-\left( 1-\lambda \right) \left( y-x\right) \frac{\varphi \left( \left( 1-\lambda \right) x+\lambda y\right) }{\Phi ^{2}\left( \left( 1-\lambda \right) x+\lambda y\right) }. \notag \end{align}
</div>
<p>If we multiply (<a href="#e.3.4.a">2.17</a>) by \(1-\lambda \) and (<a href="#e.3.4.b">2.18</a>) by \(\lambda \) and add the obtained results, we get</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \frac{1-\lambda }{\Phi \left( x\right) }+\frac{\lambda }{\Phi \left( y\right) }-\frac{1}{\Phi \left( \left( 1-\lambda \right) x+\lambda y\right) }\geq \left( \leq \right) 0 \end{equation*}
</div>
<p>for any \(\lambda \in \left[ 0,2\right] ,\) which shows that \(\Phi \) is \(AH\)-convex (concave) on \(I.\) <div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="c.3.1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">2.3</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(\Phi :I\rightarrow \left( 0,\infty \right) \) be differentiable on \(\mathring {I}.\) Then \(\Phi \) is \(AH\)-convex (concave) on \(I \) if and only if for any \(x,y\in \mathring {I}\), we have </p>
<div class="equation" id="e.3.5">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\Phi \left( y\right) }{\Phi \left( x\right) }-1\leq \left( \geq \right) \frac{\Phi ^{\prime }\left( y\right) }{\Phi \left( y\right) }\left( y-x\right) . \label{e.3.5} \end{equation}
  </div>
  <span class="equation_label">2.19</span>
</p>
</div>

  </div>
</div> </p>
<p>If \(\Delta :I\rightarrow \mathbb {R}\) is differentiable concave function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R},\) \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), and \(f:\Omega \rightarrow \left[ m,M\right] \) then by taking \(\Phi =-\Delta \) in (<a href="#b.19">1.8</a>) we get </p>
<div class="displaymath" id="e.3.6">
  \begin{align}  0&  \leq \Delta \left( \overline{f}_{\Omega ,w}\right) -\int _{\Omega }\left( \Delta \circ f\right) wd\mu \label{e.3.6} \\ &  \leq \overline{f}_{\Omega ,w}\int _{\Omega }\left( \Delta ^{\prime }\circ f\right) wd\mu -\int _{\Omega }\left( \Delta ^{\prime }\circ f\right) fwd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \Delta ^{\prime }\left( m\right) -\Delta ^{\prime }\left( M\right) \right] D_{w}\left( f\right) \notag \\ &  \leq \tfrac {1}{2}\left[ \Delta ^{\prime }\left( m\right) -\Delta ^{\prime }\left( M\right) \right] D_{w,2}\left( f\right) \notag \\ &  \leq \tfrac {1}{4}\left[ \Delta ^{\prime }\left( m\right) -\Delta ^{\prime }\left( M\right) \right] \left( M-m\right) . \notag \end{align}
</div>
<p>If \(\Phi :I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex function on the interval of real numbers \(I\) and \(m,M\in \mathbb {R},\) \(m{\lt}M\) with \(\left[ m,M\right] \subset \mathring {I}\), then by taking \(\Delta =\frac{1}{\Phi },\) which is differentiable concave, in (<a href="#e.3.6">2.20</a>) we get</p>
<div class="displaymath" id="e.3.7">
  \begin{align}  0&  \leq \left[ \Phi \left( \overline{f}_{\Omega ,w}\right) \right] ^{-1}-\int _{\Omega }\tfrac {wd\mu }{\Phi \circ f} \label{e.3.7} \\ &  \leq \int _{\Omega }\tfrac {\Phi ^{\prime }\circ f}{\Phi ^{2}\left( f\right) }fwd\mu -\overline{f}_{\Omega ,w}\int _{\Omega }\tfrac {\Phi ^{\prime }\circ f}{\Phi ^{2}\left( f\right) }wd\mu \notag \\ &  \leq \tfrac {1}{2}\left[ \tfrac {\Phi ^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\tfrac {\Phi ^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] D_{w}\left( f\right) \notag \\ &  \leq \tfrac {1}{2}\left[ \tfrac {\Phi ^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\tfrac {\Phi ^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] D_{w,2}\left( f\right) \notag \\ &  \leq \tfrac {1}{4}\left[ \tfrac {\Phi ^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\tfrac {\Phi ^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) , \notag \end{align}
</div>
<p>provided \(f:\Omega \rightarrow \left[ m,M\right] \) so that \(\frac{1}{\Phi \circ f},\) \(f,\) \(\frac{\Phi ^{\prime }\circ f}{\Phi ^{2}\left( f\right) }f,\; \frac{\Phi ^{\prime }\circ f}{\Phi ^{2}\left( f\right) }\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>If \(w\geq 0\) a.e. on \(\Omega \) with \(\int _{\Omega }wd\mu =1\) and \(\Phi :I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex on \(I\) then </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  p:=\frac{\left( \Phi \circ f\right) w}{\int _{\Omega }\left( \Phi \circ f\right) wd\mu } \end{equation*}
</div>
<p>is a weight with \(\int _{\Omega }pd\mu =1.\) </p>
<p>We have</p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \overline{f}_{\Omega ,p}=\int _{\Omega }pfd\mu =\frac{\int _{\Omega }\left( \Phi \circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi \circ f\right) wd\mu } \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \int _{\Omega }\frac{pd\mu }{\Phi \circ f}=\frac{1}{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }. \end{equation*}
</div>
<p>If we use the first inequality in (<a href="#e.3.7">2.21</a>) for the weight \(p\) we get</p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  0\leq \left[ \Phi \left( \frac{\int _{\Omega }\left( \Phi \circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }\right) \right] ^{-1}-\frac{1}{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }, \end{equation*}
</div>
<p>which is equivalent to </p>
<div class="equation" id="e.3.8">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{\Omega }\left( \Phi \circ f\right) wd\mu \geq \Phi \left( \frac{\int _{\Omega }\left( \Phi \circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }\right) , \label{e.3.8} \end{equation}
  </div>
  <span class="equation_label">2.22</span>
</p>
</div>
<p>where \(f:\Omega \rightarrow I\) and \(\Phi :I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex so that \(\left( \Phi \circ f\right) ,\) \(f,\) \(\left( \Phi \circ f\right) f,\; \left( \Phi \circ f\right) \in L_{w}\left( \Omega ,\mu \right) \). </p>
<p>Now, if we use the inequality between the first and last term in (<a href="#e.3.7">2.21</a>) we get</p>
<div class="displaymath" id="e.3.9">
  \begin{align}  0&  \leq \left[ \Phi \left( \frac{\int _{\Omega }\left( \Phi \circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }\right) \right] ^{-1}-\frac{1}{\int _{\Omega }\left( \Phi \circ f\right) wd\mu } \label{e.3.9} \\ &  \leq \frac{1}{4}\left[ \frac{\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\frac{\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) , \notag \end{align}
</div>
<p>where \(f:\Omega \rightarrow \left[ m,M\right] \subset I\) and \(\Phi :I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex so that \(\left( \Phi \circ f\right) ,\) \(f,\) \(\left( \Phi \circ f\right) f,\; \left( \Phi \circ f\right) \in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>Similar results may be stated by using various reverses of Jensen’s inequality as stated in the introduction. </p>
<p>Let \(\Phi :\left[ m,M\right] \rightarrow \mathbb {R}\) be a \(AH\)-convex function on \(\left[ m,M\right] .\) If \(x_{i}\in \left[ m,M\right] \) and \(w_{i}\geq 0\) \(\left( i=1,\dots ,n\right) \) with \(W_{n}:=\sum _{i=1}^{n}w_{i}=1,\) then from (<a href="#e.3.8">2.22</a>) one has the weighted discrete inequality: </p>
<div class="equation" id="e.3.9.a">
<p>
  <div class="equation_content">
    \begin{equation}  \Phi \left( \tfrac {\sum _{i=1}^{n}w_{i}x_{i}\Phi \left( x_{i}\right) }{\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) }\right) \leq \sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) \label{e.3.9.a} \end{equation}
  </div>
  <span class="equation_label">2.24</span>
</p>
</div>
<p>while from (<a href="#e.3.9">2.23</a>) we have</p>
<div class="displaymath" id="e.3.9.b">
  \begin{align}  0&  \leq \left[ \Phi \left( \tfrac {\sum _{i=1}^{n}w_{i}x_{i}\Phi \left( x_{i}\right) }{\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) }\right) \right] ^{-1}-\tfrac {1}{\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) } \label{e.3.9.b} \\ &  \leq \tfrac {1}{4}\left[ \tfrac {\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\tfrac {\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) . \notag \end{align}
</div>
<h1 id="a0000000025">3 New Results</h1>
<p>In what follows, we assume that \(w:\Omega \rightarrow \mathbb {R}\), with \(w\left( x\right) \geq 0\) for \(\mu \)-a.e. \(x\in \Omega ,\) is a \(\mu \)-measurable function with \(\int _{\Omega }wd\mu =1.\) </p>
<p>The following result holds: </p>
<p><div class="theorem_thmwrapper " id="t.4.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\Phi :I\rightarrow \left( 0,\infty \right) \) be differentiable on \(\mathring {I}.\) If \(\Phi \) is \(AH\)-convex (concave) on \(I\) and \(f:\Omega \rightarrow I\) so that \(\Phi \circ f,\) \(\frac{\left( \Phi ^{\prime }\circ f\right) f}{\Phi \circ f},\; \frac{\Phi ^{\prime }\circ f}{\Phi \circ f}\in L_{w}\left( \Omega ,\mu \right) ,\) then</p>
<div class="displaymath" id="e.4.1">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\int _{\Omega }\left( \Phi \circ f\right) wd\mu -1 \leq \left( \geq \right) \label{e.4.1} \\ &  \leq \left( \geq \right) \int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu -x\int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu \notag \end{align}
</div>
<p>for any \(x\in I.\) </p>
<p>Moreover, if</p>
<div class="equation" id="e.4.2">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu }{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu }\in I, \label{e.4.2} \end{equation}
  </div>
  <span class="equation_label">3.27</span>
</p>
</div>
<p>then</p>
<div class="equation" id="e.4.3">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{\Omega }\left( \Phi \circ f\right) wd\mu \leq \left( \geq \right) \Phi \left( \frac{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu }{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu }\right) . \label{e.4.3} \end{equation}
  </div>
  <span class="equation_label">3.28</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> From (<a href="#e.3.5">2.19</a>) we have for any \(x\in I\), that </p>
<div class="equation" id="e.3.10">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\Phi \circ f}{\Phi \left( x\right) }-1\leq \left( \geq \right) \tfrac {\Phi ^{\prime }\circ f}{\Phi \circ f}\left( f-x\right) \label{e.3.10} \end{equation}
  </div>
  <span class="equation_label">3.29</span>
</p>
</div>
<p>almost everywhere on \(\Omega .\) </p>
<p>If we multiply this by \(w\geq 0\) a.e. on \(\Omega \) we get</p>
<div class="equation" id="e.3.11">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\left( \Phi \circ f\right) w}{\Phi \left( x\right) }-w\leq \left( \geq \right) \tfrac {\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}-\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}x, \label{e.3.11} \end{equation}
  </div>
  <span class="equation_label">3.30</span>
</p>
</div>
<p>almost everywhere on \(\Omega .\) </p>
<p>Integrating the inequality (<a href="#e.3.11">3.30</a>) on \(\Omega \) and taking into account that \(\int _{\Omega }wd\mu =1,\) we get</p>
<div class="displaymath" id="e.3.12">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\int _{\Omega }\left( \Phi \circ f\right) wd\mu -1 \leq \left( \geq \right) \label{e.3.12} \\ &  \leq \left( \geq \right) \int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu -x\int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu \notag \end{align}
</div>
<p>and the inequality (<a href="#e.4.1">3.26</a>) is proved. </p>
<p>The inequality (<a href="#e.4.3">3.28</a>) follows by (<a href="#e.4.1">3.26</a>) by taking </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  x=\frac{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu }{\int _{\Omega }\frac{\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu }\in I. \end{equation*}
</div>
<p> <div class="proof_wrapper" id="a0000000028">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="c.4.1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3.2</span>
  </div>
  <div class="corollary_thmcontent">
  <p>With the assumptions of Theorem <a href="#t.4.1">3.1</a> and if \(f:\Omega \rightarrow \left[ m,M\right] \subset I\) and \(\Phi \) is monotonic nondecreasing or nonincreasing on \(\left[ m,M\right] ,\) then the inequality (<a href="#e.4.3">3.28</a>) holds. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000029">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(m\leq f\leq M\) a.e. on \(\Omega ,\) then for \(\Phi \) monotonic nondecreasing we have</p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  m\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}\leq \tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}f\leq M\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f} \end{equation*}
</div>
<p>a.e. on \(\Omega .\) </p>
<p>Integrating on \(\Omega \) we get</p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  m\int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu \leq \int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}fd\mu \leq M\int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu , \end{equation*}
</div>
<p>which shows that the condition (<a href="#e.4.2">3.27</a>) is satisfied. </p>
<p>The case of \(\Phi \) is nonincreasing goes likewise and the statement is proved. <div class="proof_wrapper" id="a0000000032">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="r.4.1">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(f:\Omega \rightarrow \left[ m,M\right] \subset I\), then by taking </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  x=\int _{\Omega }fwd\mu \in \left[ m,M\right] \end{equation*}
</div>
<p>in (<a href="#e.4.1">3.26</a>) we get the inequality</p>
<div class="displaymath" id="e.3.12.a">
  \begin{align} &  \tfrac {1}{\Phi \left( \int _{\Omega }fwd\mu \right) }\int _{\Omega }\left( \Phi \circ f\right) wd\mu -1 \leq \left( \geq \right) \label{e.3.12.a} \\ &  \leq \left( \geq \right) \int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) fw}{\Phi \circ f}d\mu -\int _{\Omega }fwd\mu \int _{\Omega }\tfrac {\left( \Phi ^{\prime }\circ f\right) w}{\Phi \circ f}d\mu . \notag \end{align}
</div>

  </div>
</div> </p>
<p>Let \(\Phi :\left[ m,M\right] \rightarrow \mathbb {R}\) be a differentiable \(AH\)-convex (concave) function on \(\left[ m,M\right] .\) If \(x_{i}\in \left[ m,M\right] \) and \(w_{i}\geq 0\) \(\left( i=1,\dots ,n\right) \) with \(W_{n}:=\sum _{i=1}^{n}w_{i}=1,\) then from (<a href="#e.4.1">3.26</a>) one has the weighted discrete inequality:</p>
<div class="displaymath" id="e.3.12.a.1">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) -1 \leq \left( \geq \right) \label{e.3.12.a.1} \\ &  \leq \left( \geq \right) \sum _{i=1}^{n}\tfrac {\Phi ^{\prime }\left( x_{i}\right) x_{i}w_{i}}{\Phi \left( x_{i}\right) }-x\sum _{i=1}^{n}\tfrac {\Phi ^{\prime }\left( x_{i}\right) w_{i}}{\Phi \left( x_{i}\right) } \notag \end{align}
</div>
<p>for any \(x\in \left[ m,M\right] .\) </p>
<p>In particular, we have </p>
<div class="displaymath" id="e.3.12.a.2">
  \begin{align} &  \tfrac {1}{\Phi \left( \sum _{i=1}^{n}w_{i}x_{i}\right) }\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) -1 \leq \left( \geq \right) \label{e.3.12.a.2} \\ &  \leq \left( \geq \right) \sum _{i=1}^{n}\tfrac {\Phi ^{\prime }\left( x_{i}\right) x_{i}w_{i}}{\Phi \left( x_{i}\right) }-\sum _{i=1}^{n}w_{i}x_{i}\sum _{i=1}^{n}\tfrac {\Phi ^{\prime }\left( x_{i}\right) w_{i}}{\Phi \left( x_{i}\right) }. \notag \end{align}
</div>
<p>Moreover, if \(\Phi \) is monotonic nondecreasing or nonincreasing on \(\left[ m,M\right] ,\) then we have the inequality</p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  \sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) \leq \left( \geq \right) \Phi \left( \frac{\sum _{i=1}^{n}\frac{\Phi ^{\prime }\left( x_{i}\right) x_{i}w_{i}}{\Phi \left( x_{i}\right) }}{\sum _{i=1}^{n}\frac{\Phi ^{\prime }\left( x_{i}\right) w_{i}}{\Phi \left( x_{i}\right) }}\right) . \end{equation*}
</div>
<p>The following result also holds: </p>
<p><div class="theorem_thmwrapper " id="t.4.2.a">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\Phi :I\rightarrow \left( 0,\infty \right) \) be differentiable on \(\mathring {I}.\) If \(\Phi \) is \(AH\)-convex (concave) on \(I\) and \(f:\Omega \rightarrow I\) so that \(\Phi \circ f,\) \(\frac{\left( \Phi ^{\prime }\circ f\right) f}{\Phi \circ f},\; \frac{\Phi ^{\prime }\circ f}{\Phi \circ f}\in L_{w}\left( \Omega ,\mu \right) ,\) then</p>
<div class="displaymath" id="e.3.13">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\int _{\Omega }\left( \Phi \circ f\right) ^{2}wd\mu -\int _{\Omega }\left( \Phi \circ f\right) wd\mu \leq \left( \geq \right) \label{e.3.13} \\ &  \leq \left( \geq \right) \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu -x\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu \notag \end{align}
</div>
<p>for any \(x\in I.\) </p>
<p>Moreover, if</p>
<div class="equation" id="e.3.14">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu }\in I, \label{e.3.14} \end{equation}
  </div>
  <span class="equation_label">3.36</span>
</p>
</div>
<p>then</p>
<div class="equation" id="e.3.15">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\int _{\Omega }\left( \Phi \circ f\right) ^{2}wd\mu }{\int _{\Omega }\left( \Phi \circ f\right) wd\mu }\leq \left( \geq \right) \Phi \left( \tfrac {\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu }\right) . \label{e.3.15} \end{equation}
  </div>
  <span class="equation_label">3.37</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000035">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> From (<a href="#e.3.5">2.19</a>) we have for any \(x\in I\), that </p>
<div class="displaymath" id="a0000000036">
  \begin{equation*}  \tfrac {1}{\Phi \left( x\right) }\left( \Phi \circ f\right) ^{2}-\Phi \circ f\leq \left( \geq \right) \left( \Phi ^{\prime }\circ f\right) \left( f-x\right) \end{equation*}
</div>
<p>almost everywhere on \(\Omega .\) </p>
<p>If we multiply this by \(w\geq 0\) a.e. on \(\Omega \) we get</p>
<div class="equation" id="e.3.16">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {1}{\Phi \left( x\right) }\left( \Phi \circ f\right) ^{2}w-\left( \Phi \circ f\right) w\leq \left( \geq \right) \left( \Phi ^{\prime }\circ f\right) fw-\left( \Phi ^{\prime }\circ f\right) wx \label{e.3.16} \end{equation}
  </div>
  <span class="equation_label">3.38</span>
</p>
</div>
<p>almost everywhere on \(\Omega .\) </p>
<p>Integrating the inequality (<a href="#e.3.16">3.38</a>) on \(\Omega \) and taking into account that \(\int _{\Omega }wd\mu =1,\) we get</p>
<div class="displaymath" id="e.3.17">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\int _{\Omega }\left( \Phi \circ f\right) ^{2}wd\mu -\int _{\Omega }\left( \Phi \circ f\right) wd\mu \leq \left( \geq \right) \label{e.3.17} \\ &  \leq \left( \geq \right) \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu -x\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu \notag \end{align}
</div>
<p>and the inequality (<a href="#e.4.1">3.26</a>) is proved. </p>
<p>If we take </p>
<div class="displaymath" id="a0000000037">
  \begin{equation*}  x=\tfrac {\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu }{\int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu } \end{equation*}
</div>
<p>in (<a href="#e.3.13">3.35</a>) and do the required calculation, we get the desired result (<a href="#e.3.15">3.37</a>). <div class="proof_wrapper" id="a0000000038">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="c.4.2">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3.5</span>
  </div>
  <div class="corollary_thmcontent">
  <p>With the assumptions of Theorem <a href="#t.4.2.a">3.4</a> and if \(f:\Omega \rightarrow \left[ m,M\right] \subset I\) and \(\Phi \) is monotonic nondecreasing (nonincreasing) on \(\left[ m,M\right] ,\) then the inequality (<a href="#e.3.15">3.37</a>) holds. </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="r.4.2">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.6</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(f:\Omega \rightarrow \left[ m,M\right] \subset I\), then by taking </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  x=\int _{\Omega }fwd\mu \in \left[ m,M\right] \end{equation*}
</div>
<p>in (<a href="#e.3.13">3.35</a>) we get the inequality</p>
<div class="displaymath" id="e.3.17.a">
  \begin{align} &  \tfrac {1}{\Phi \left( \int _{\Omega }fwd\mu \right) }\int _{\Omega }\left( \Phi \circ f\right) ^{2}wd\mu -\int _{\Omega }\left( \Phi \circ f\right) wd\mu \leq \left( \geq \right) \label{e.3.17.a} \\ &  \leq \left( \geq \right) \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) fwd\mu -\int _{\Omega }fwd\mu \int _{\Omega }\left( \Phi ^{\prime }\circ f\right) wd\mu . \notag \end{align}
</div>

  </div>
</div> </p>
<p>Let \(\Phi :\left[ m,M\right] \rightarrow \mathbb {R}\) be a differentiable \(AH\)-convex (concave) function on \(\left[ m,M\right] .\) If \(x_{i}\in \left[ m,M\right] \) and \(w_{i}\geq 0\) \(\left( i=1,\dots ,n\right) \) with \(W_{n}:=\sum _{i=1}^{n}w_{i}=1,\) then from (<a href="#e.3.13">3.35</a>) one has the weighted discrete inequality:</p>
<div class="displaymath" id="e.3.18">
  \begin{align} &  \tfrac {1}{\Phi \left( x\right) }\sum _{i=1}^{n}w_{i}\Phi ^{2}\left( x_{i}\right) -\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) \leq \left( \geq \right) \label{e.3.18} \\ &  \leq \left( \geq \right) \sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) x_{i}-x\sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) \notag \end{align}
</div>
<p>for any \(x\in \left[ m,M\right] .\) </p>
<p>In particular we have</p>
<div class="displaymath" id="e.3.19">
  \begin{align} &  \tfrac {1}{\Phi \left( \sum _{i=1}^{n}w_{i}x_{i}\right) }\sum _{i=1}^{n}w_{i}\Phi ^{2}\left( x_{i}\right) -\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) \leq \left( \geq \right) \label{e.3.19} \\ &  \leq \left( \geq \right) \sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) x_{i}-\sum _{i=1}^{n}w_{i}x_{i}\sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) . \notag \end{align}
</div>
<p>Moreover, if \(\Phi \) is monotonic nondecreasing or nonincreasing on \(\left[ m,M\right] ,\) then we have the inequality</p>
<div class="equation" id="e.3.20">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\sum _{i=1}^{n}w_{i}\Phi ^{2}\left( x_{i}\right) }{\sum _{i=1}^{n}w_{i}\Phi \left( x_{i}\right) }\leq \left( \geq \right) \Phi \left( \tfrac {\sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) x_{i}}{\sum _{i=1}^{n}w_{i}\Phi ^{\prime }\left( x_{i}\right) }\right) . \label{e.3.20} \end{equation}
  </div>
  <span class="equation_label">3.43</span>
</p>
</div>
<h1 id="a0000000040">4 Some Examples</h1>
<p>Consider the function</p>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  \Phi _{p}\left( t\right) =t^{p}=\tfrac {1}{t^{-p}} \end{equation*}
</div>
<p>if \(-p{\gt}1\) or \(-p{\lt}0,\) i.e. \(p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \) then the function \(\Phi _{p}\left( t\right) =t^{p},t{\gt}0\) is \(AM\)-concave. If \(p\in \left( -1,0\right) \) then the function \(\Phi _{p}\left( t\right) =t^{p},t{\gt}0\) is \(AM\)-convex. </p>
<p>If we apply the inequality (<a href="#e.4.3">3.28</a>) for \(\Phi _{p},\) then we get </p>
<div class="equation" id="e.5.1">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{\Omega }f^{p}wd\mu \left( \int _{\Omega }\tfrac {w}{f}d\mu \right) ^{p}\leq \left( \geq \right) 1, \label{e.5.1} \end{equation}
  </div>
  <span class="equation_label">4.44</span>
</p>
</div>
<p>where \(p\in \left( -1,0\right) \left( p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \right) ,\) and \(f:\Omega \rightarrow \left( 0,\infty \right) \) is such that \(f^{p},\) \(\frac{1}{f}\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>If we use the inequality (<a href="#e.3.15">3.37</a>) for \(\Phi _{p},\) then we get</p>
<div class="equation" id="e.5.2">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\int _{\Omega }f^{2p}wd\mu }{\int _{\Omega }f^{p}wd\mu }\leq \left( \geq \right) \left( \tfrac {\int _{\Omega }f^{p}wd\mu }{\int _{\Omega }f^{p-1}wd\mu }\right) ^{p}, \label{e.5.2} \end{equation}
  </div>
  <span class="equation_label">4.45</span>
</p>
</div>
<p>where \(p\in \left( -1,0\right) \left( p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \right) ,\) and \(f:\Omega \rightarrow \left( 0,\infty \right) \) is such that \(f^{2p},\) \(f^{p},\) \(f^{p-1}\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>If we use the inequality (<a href="#e.3.12.a">3.32</a>) for \(\Phi _{p},\) then we have</p>
<div class="equation" id="e.5.2.a">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{1}{\left( \int _{\Omega }fwd\mu \right) ^{p}}\int _{\Omega }f^{p}wd\mu -1\leq \left( \geq \right) p\left[ 1-\int _{\Omega }fwd\mu \int _{\Omega }\tfrac {w}{f}d\mu \right] , \label{e.5.2.a} \end{equation}
  </div>
  <span class="equation_label">4.46</span>
</p>
</div>
<p>where \(p\in \left( -1,0\right) \left( p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \right) ,\) and \(f:\Omega \rightarrow \left( 0,\infty \right) \) is such that \(f^{p},\) \(f,\) \(\frac{1}{f}\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>By Čebyšev inequality for asynchronous functions we have </p>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  1\leq \int _{\Omega }fwd\mu \int _{\Omega }\tfrac {w}{f}d\mu \end{equation*}
</div>
<p>and then by (<a href="#e.5.2">4.45</a>) for \(p\in \left( -\infty ,-1\right) \) we have </p>
<div class="equation" id="e.5.2.a.1">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{1}{\left( \int _{\Omega }fwd\mu \right) ^{p}}\int _{\Omega }f^{p}wd\mu -1\geq p\left[ 1-\int _{\Omega }fwd\mu \int _{\Omega }\tfrac {w}{f}d\mu \right] \geq 0. \label{e.5.2.a.1} \end{equation}
  </div>
  <span class="equation_label">4.47</span>
</p>
</div>
<p>If we denote \(r=-p\in \left( 1,\infty \right) ,\) then the inequality (<a href="#e.5.2.a.1">4.47</a>) can be written as</p>
<div class="equation" id="e.5.2.a.2">
<p>
  <div class="equation_content">
    \begin{equation}  \left( \int _{\Omega }fwd\mu \right) ^{r}\int _{\Omega }\tfrac {w}{f^{r}}d\mu -1\geq r\left[ \int _{\Omega }fwd\mu \int _{\Omega }\tfrac {w}{f}d\mu -1\right] \geq 0 \label{e.5.2.a.2} \end{equation}
  </div>
  <span class="equation_label">4.48</span>
</p>
</div>
<p>provided \(\frac{1}{f^{r}},\) \(f,\) \(\frac{1}{f}\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>Also, if we use the inequality (<a href="#e.3.17.a">3.40</a>) for \(\Phi _{p},\) then we have </p>
<div class="displaymath" id="e.5.2.b">
  \begin{align} &  \frac{1}{\left( \int _{\Omega }fwd\mu \right) ^{p}}\int _{\Omega }f^{2p}wd\mu -\int _{\Omega }f^{p}wd\mu \leq \left( \geq \right) \label{e.5.2.b} \\ &  \leq \left( \geq \right) p\left[ \int _{\Omega }f^{p}wd\mu -\int _{\Omega }fwd\mu \int _{\Omega }f^{p-1}wd\mu \right] , \notag \end{align}
</div>
<p>where \(p\in \left( -1,0\right) \left( p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \right) ,\) and \(f:\Omega \rightarrow \left( 0,\infty \right) \) is such that \(f^{2p},\) \(f^{p},\) \(f^{p-1},\) \(f\in L_{w}\left( \Omega ,\mu \right) .\) </p>
<p>We observe that by Čebyšev inequality for synchronous functions we have for \(p\geq 1\)</p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \int _{\Omega }f^{p}wd\mu \geq \int _{\Omega }fwd\mu \int _{\Omega }f^{p-1}wd\mu \end{equation*}
</div>
<p>and by (<a href="#e.5.2.b">4.49</a>) we have</p>
<div class="displaymath" id="e.5.2.b.1">
  \begin{align} &  \frac{1}{\left( \int _{\Omega }fwd\mu \right) ^{p}}\int _{\Omega }f^{2p}wd\mu -\int _{\Omega }f^{p}wd\mu \geq \label{e.5.2.b.1} \\ &  \geq p\left[ \int _{\Omega }f^{p}wd\mu -\int _{\Omega }fwd\mu \int _{\Omega }f^{p-1}wd\mu \right] \geq 0 \notag \end{align}
</div>
<p>for \(p\geq 1.\) </p>
<p>Now consider the function \(\Phi _{\ln }\left( t\right) =\ln t,\) \(t{\gt}1.\) The function</p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  g\left( t\right) :=\tfrac {1}{\ln t},\text{ }t{\gt}1 \end{equation*}
</div>
<p>is convex on \(\left( 1,\infty \right) .\) Therefore \(\Phi _{\ln }\left( t\right) =\ln t,\) \(t{\gt}1\) is a \(AM\)-concave function on \(\left( 1,\infty \right) .\) </p>
<p>Let \(f:\Phi \rightarrow \left( 1,\infty \right) \) be so that \(\ln f,\) \(\frac{1}{\ln f},\) \(\frac{1}{f\ln f}\in L_{w}\left( \Omega ,\mu \right) ,\) then by using the inequality (<a href="#e.4.3">3.28</a>) for \(\Phi _{\ln }\) we have </p>
<div class="equation" id="e.5.3">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{\Omega }w\ln fd\mu \geq \ln \left( \tfrac {\int _{\Omega }\frac{w}{\ln f}d\mu }{\int _{\Omega }\frac{w}{f\ln f}d\mu }\right) . \label{e.5.3} \end{equation}
  </div>
  <span class="equation_label">4.51</span>
</p>
</div>
<p>Let \(f:\Phi \rightarrow \left( 1,\infty \right) \) be so that \(\left( \ln f\right) ^{2},\) \(\ln f,\) \(\frac{1}{f}\in L_{w}\left( \Omega ,\mu \right) \), then by (<a href="#e.3.15">3.37</a>) for \(\Phi _{\ln }\) we get</p>
<div class="equation" id="e.5.4">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\int _{\Omega }w\left( \ln f\right) ^{2}d\mu }{\int _{\Omega }w\ln fd\mu }\geq \ln \left( \tfrac {1}{\int _{\Omega }\frac{w}{f}d\mu }\right) . \label{e.5.4} \end{equation}
  </div>
  <span class="equation_label">4.52</span>
</p>
</div>
<h1 id="a0000000045">5 Applications for Functions of Selfadjoint Operators</h1>
<p>Let \(A\) be a selfadjoint operator on the complex Hilbert space \(\left( H,\left\langle .,.\right\rangle \right) \) with the spectrum \(Sp\left( A\right) \) included in the interval \(\left[ m,M\right] \) for some real numbers \(m{\lt}M\) and let \(\left\{  E_{\lambda }\right\}  _{\lambda }\) be its <i class="itshape">spectral family</i>. Then for any continuous function \(f:\left[ m,M\right] \rightarrow \left[ a,b\right] \), it is well known that we have the following <i class="itshape">spectral representation in terms of the Riemann-Stieltjes integral </i>(see for instance <span class="cite">
	[
	<a href="#H" >13</a>
	, 
	p. 257
	]
</span>): </p>
<div class="equation" id="e.6.1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle f\left( A\right) x,y\right\rangle =\int _{m-0}^{M}f\left( \lambda \right) d\left( \left\langle E_{\lambda }x,y\right\rangle \right) , \label{e.6.1} \end{equation}
  </div>
  <span class="equation_label">5.53</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert f\left( A\right) x\right\Vert ^{2}=\int _{m-0}^{M}\left\vert f\left( \lambda \right) \right\vert ^{2}d\left\Vert E_{\lambda }x\right\Vert ^{2}, \label{e.6.2} \end{equation}
  </div>
  <span class="equation_label">5.54</span>
</p>
</div>
<p>for any \(x,y\in H.\) </p>
<p>The function \(g_{x,y}\left( \lambda \right) :=\left\langle E_{\lambda }x,y\right\rangle \) is of <i class="itshape">bounded variation</i> on the interval \(\left[ m,M\right] \) and </p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  g_{x,y}\left( m-0\right) =0\text{ while }g_{x,y}\left( M\right) =\left\langle x,y\right\rangle \end{equation*}
</div>
<p>for any \(x,y\in H.\) It is also well known that \(g_{x}\left( \lambda \right) :=\left\langle E_{\lambda }x,x\right\rangle \) is <i class="itshape">monotonic nondecreasing</i> and <i class="itshape">right continuous</i> on \(\left[ m,M\right] \) for any \(x\in H\). </p>
<p>Now, assume that \(\Phi :\left[ a,b\right] \subset I\rightarrow \left( 0,\infty \right) \) is continuous \(AH\)-convex function on the interval of real numbers \(I,\) \(f:\left[ m,M\right] \rightarrow \left[ a,b\right] ,\) \(p:\left[ m,M\right] \rightarrow (0,\infty )\) are continuous functions on \(\left[ m,M\right] \) and \(g:\left[ m,M\right] \rightarrow \mathbb {R}\) is monotonic nondecreasing on \(\left[ m,M\right] .\) </p>
<p>By (<a href="#e.3.8">2.22</a>) and (<a href="#e.3.9">2.23</a>) we have the following inequalities for the Riemann-Stieltjes integral:</p>
<div class="equation" id="e.6.3">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }{\int _{m}^{M}p\left( t\right) dg\left( t\right) }\geq \Phi \left( \frac{\int _{m}^{M}\Phi \left( f\left( t\right) \right) f\left( t\right) p\left( t\right) dg\left( t\right) }{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }\right) , \label{e.6.3} \end{equation}
  </div>
  <span class="equation_label">5.55</span>
</p>
</div>
<p>and</p>
<div class="displaymath" id="e.6.4">
  \begin{align}  0&  \leq \left[ \Phi \left( \tfrac {\int _{m}^{M}\Phi \left( f\left( t\right) \right) f\left( t\right) p\left( t\right) dg\left( t\right) }{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }\right) \right] ^{-1}-\frac{\int _{m}^{M}p\left( t\right) dg\left( t\right) }{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) } \label{e.6.4} \\ &  \leq \tfrac {1}{4}\left[ \frac{\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\frac{\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) . \notag \end{align}
</div>
<p>Now, if we apply the inequalities (<a href="#e.6.3">5.55</a>) and (<a href="#e.6.4">5.56</a>) for the monotonic nondecreasing function \(g_{x}\left( \lambda \right) :=\left\langle E_{\lambda }x,x\right\rangle ,\) \(x\in H,\) where \(\left\{  E_{\lambda }\right\}  _{\lambda }\) is the spectral family of \(A,\) then we get</p>
<div class="equation" id="e.6.5">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }{\left\langle p\left( A\right) x,x\right\rangle }\geq \Phi \left( \frac{\left\langle \Phi \left( f\left( A\right) \right) f\left( A\right) p\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }\right) , \label{e.6.5} \end{equation}
  </div>
  <span class="equation_label">5.57</span>
</p>
</div>
<p>and</p>
<div class="displaymath" id="e.6.6">
  \begin{align}  0&  \leq \left[ \Phi \left( \frac{\left\langle \Phi \left( f\left( A\right) \right) f\left( A\right) p\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }\right) \right] ^{-1}-\frac{\left\langle p\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle } \label{e.6.6} \\ &  \leq \tfrac {1}{4}\left[ \frac{\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\frac{\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) , \notag \end{align}
</div>
<p>for any \(x\in H,\) \(x\neq 0.\) </p>
<p>In particular, if \(p\) is taken to be the constant \(1,\) then for any \(x\in H,\) \(\left\Vert x\right\Vert =1,\) we have</p>
<div class="equation" id="e.6.7">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle \geq \Phi \left( \frac{\left\langle \Phi \left( f\left( A\right) \right) f\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle }\right) , \label{e.6.7} \end{equation}
  </div>
  <span class="equation_label">5.59</span>
</p>
</div>
<p>and</p>
<div class="displaymath" id="e.6.8">
  \begin{align}  0&  \leq \left[ \Phi \left( \frac{\left\langle \Phi \left( f\left( A\right) \right) f\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle }\right) \right] ^{-1}-\frac{1}{\left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle } \label{e.6.8} \\ &  \leq \tfrac {1}{4}\left[ \frac{\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\frac{\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) . \notag \end{align}
</div>
<p>Moreover, if \(\left[ a,b\right] =\left[ m,M\right] \) and \(f\left( t\right) =t,\) then from (<a href="#e.6.7">5.59</a>) and (<a href="#e.6.8">5.60</a>) we get</p>
<div class="equation" id="e.6.9">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \Phi \left( A\right) x,x\right\rangle \geq \Phi \left( \frac{\left\langle \Phi \left( A\right) Ax,x\right\rangle }{\left\langle \Phi \left( A\right) x,x\right\rangle }\right) , \label{e.6.9} \end{equation}
  </div>
  <span class="equation_label">5.61</span>
</p>
</div>
<p>and</p>
<div class="displaymath" id="e.6.10">
  \begin{align}  0&  \leq \left[ \Phi \left( \frac{\left\langle \Phi \left( A\right) Ax,x\right\rangle }{\left\langle \Phi \left( A\right) x,x\right\rangle }\right) \right] ^{-1}-\frac{1}{\left\langle \Phi \left( A\right) x,x\right\rangle } \label{e.6.10} \\ &  \leq \tfrac {1}{4}\left[ \frac{\Phi _{-}^{\prime }\left( M\right) }{\Phi ^{2}\left( M\right) }-\frac{\Phi _{+}^{\prime }\left( m\right) }{\Phi ^{2}\left( m\right) }\right] \left( M-m\right) . \notag \end{align}
</div>
<p>for any \(x\in H,\) \(\left\Vert x\right\Vert =1.\) </p>
<p>Further on, assume that \(\Phi :\left[ a,b\right] \subset I\rightarrow \left( 0,\infty \right) \) is continuously differentiable \(AH\)-convex and monotonic nondecreasing (nonincreasing) function on the interval of real numbers \(I,\) \(f:\left[ m,M\right] \rightarrow \left[ a,b\right] ,\) \(p:\left[ m,M\right] \rightarrow (0,\infty )\) are continuous functions on \(\left[ m,M\right] \) and \(g:\left[ m,M\right] \rightarrow \mathbb {R}\) is monotonic nondecreasing on \(\left[ m,M\right] .\) </p>
<p>If we write the inequalities (<a href="#e.4.3">3.28</a>) and (<a href="#e.3.15">3.37</a>) for the Riemann-Stieltjes integral, then we have </p>
<div class="equation" id="e.6.11">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }{\int _{m}^{M}p\left( t\right) dg\left( t\right) }\leq \Phi \left( \frac{\int _{m}^{M}\frac{\Phi ^{\prime }\left( f\left( t\right) \right) f\left( t\right) p\left( t\right) }{\Phi \left( f\left( t\right) \right) }dg\left( t\right) }{\int _{m}^{M}\frac{\Phi ^{\prime }\left( f\left( t\right) \right) p\left( t\right) }{\Phi \left( f\left( t\right) \right) }dg\left( t\right) }\right) \label{e.6.11} \end{equation}
  </div>
  <span class="equation_label">5.63</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.12">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\int _{m}^{M}\left( \Phi \left( f\left( t\right) \right) \right) ^{2}p\left( t\right) dg\left( t\right) }{\int _{m}^{M}\Phi \left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }\leq \Phi \left( \frac{\int _{m}^{M}\Phi ^{\prime }\left( f\left( t\right) \right) f\left( t\right) p\left( t\right) dg\left( t\right) }{\int _{m}^{M}\Phi ^{\prime }\left( f\left( t\right) \right) p\left( t\right) dg\left( t\right) }\right) . \label{e.6.12} \end{equation}
  </div>
  <span class="equation_label">5.64</span>
</p>
</div>
<p>The inequalities (<a href="#e.6.11">5.63</a>) and (<a href="#e.6.12">5.64</a>) imply the following operator inequalities </p>
<div class="equation" id="e.6.13">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }{\left\langle p\left( A\right) x,x\right\rangle }\leq \Phi \left( \frac{\left\langle \frac{\Phi ^{\prime }\left( f\left( A\right) \right) f\left( A\right) p\left( A\right) }{\Phi \left( f\left( A\right) \right) }x,x\right\rangle }{\left\langle \frac{\Phi ^{\prime }\left( f\left( A\right) \right) p\left( A\right) }{\Phi \left( f\left( A\right) \right) }x,x\right\rangle }\right) \label{e.6.13} \end{equation}
  </div>
  <span class="equation_label">5.65</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.14">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\left\langle \left( \Phi \left( f\left( A\right) \right) \right) ^{2}p\left( A\right) x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }\leq \Phi \left( \frac{\left\langle \Phi ^{\prime }\left( f\left( A\right) \right) f\left( A\right) p\left( A\right) x,x\right\rangle }{\left\langle \Phi ^{\prime }\left( f\left( A\right) \right) p\left( A\right) x,x\right\rangle }\right) \label{e.6.14} \end{equation}
  </div>
  <span class="equation_label">5.66</span>
</p>
</div>
<p>for any \(x\in H,\) \(x\neq 0.\) </p>
<p>In particular, if \(p\) is taken to be the constant \(1,\) then for any \(x\in H,\) \(\left\Vert x\right\Vert =1,\) we have </p>
<div class="equation" id="e.6.15">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle \leq \Phi \left( \frac{\left\langle \frac{\Phi ^{\prime }\left( f\left( A\right) \right) f\left( A\right) }{\Phi \left( f\left( A\right) \right) }x,x\right\rangle }{\left\langle \frac{\Phi ^{\prime }\left( f\left( A\right) \right) }{\Phi \left( f\left( A\right) \right) }x,x\right\rangle }\right) \label{e.6.15} \end{equation}
  </div>
  <span class="equation_label">5.67</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.16">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\left\langle \left( \Phi \left( f\left( A\right) \right) \right) ^{2}x,x\right\rangle }{\left\langle \Phi \left( f\left( A\right) \right) x,x\right\rangle }\leq \Phi \left( \frac{\left\langle \Phi ^{\prime }\left( f\left( A\right) \right) f\left( A\right) x,x\right\rangle }{\left\langle \Phi ^{\prime }\left( f\left( A\right) \right) x,x\right\rangle }\right) . \label{e.6.16} \end{equation}
  </div>
  <span class="equation_label">5.68</span>
</p>
</div>
<p>Moreover, if \(\left[ a,b\right] =\left[ m,M\right] \) and \(f\left( t\right) =t,\) then from (<a href="#e.6.15">5.67</a>) and (<a href="#e.6.16">5.68</a>) we get </p>
<div class="equation" id="e.6.17">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \Phi \left( A\right) x,x\right\rangle \leq \Phi \left( \frac{\left\langle \frac{\Phi ^{\prime }\left( A\right) A}{\Phi \left( A\right) }x,x\right\rangle }{\left\langle \frac{\Phi ^{\prime }\left( A\right) }{\Phi \left( A\right) }x,x\right\rangle }\right) \label{e.6.17} \end{equation}
  </div>
  <span class="equation_label">5.69</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.18">
<p>
  <div class="equation_content">
    \begin{equation}  \frac{\left\langle \left( \Phi \left( A\right) \right) ^{2}x,x\right\rangle }{\left\langle \Phi \left( A\right) x,x\right\rangle }\leq \Phi \left( \frac{\left\langle \Phi ^{\prime }\left( A\right) Ax,x\right\rangle }{\left\langle \Phi ^{\prime }\left( A\right) x,x\right\rangle }\right) , \label{e.6.18} \end{equation}
  </div>
  <span class="equation_label">5.70</span>
</p>
</div>
<p> for any \(x\in H,\) \(\left\Vert x\right\Vert =1.\) </p>
<p>If \(\Phi :\left[ a,b\right] \subset I\rightarrow \left( 0,\infty \right) \) is \(AH\)-concave, then the reverse versions of the inequalities above hold. We omit the details. </p>
<p>The interested reader may state various power inequalities for positive definite operators by choosing \(\Phi _{p}\left( t\right) =t^{p}\) which is \(AH \)-convex for \(p\in \left( -1,0\right) \) and \(AM\)-concave for \(p\in \left( -\infty ,-1\right) \cup \left( 0,\infty \right) \). </p>
<p>For instance, if we take \(p=-r,\) with \(r\in \left( 0,1\right) \) then from (<a href="#e.6.7">5.59</a>), and (<a href="#e.6.18">5.70</a>) we get</p>
<div class="displaymath" id="a0000000047">
  \begin{equation*}  \left\langle A^{-r}x,x\right\rangle \geq \left( \frac{\left\langle A^{1-r}x,x\right\rangle }{\left\langle A^{-r}x,x\right\rangle }\right) ^{-r}, \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000048">
  \begin{equation*}  \frac{\left\langle A^{-2r}x,x\right\rangle }{\left\langle A^{-r}x,x\right\rangle }\leq \left( \frac{\left\langle A^{-r}x,x\right\rangle }{\left\langle A^{-r-1}x,x\right\rangle }\right) ^{-r}, \end{equation*}
</div>
<p>which can be written as </p>
<div class="equation" id="e.6.19">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle A^{-r}x,x\right\rangle ^{1-r}\left\langle A^{1-r}x,x\right\rangle ^{r}\geq 1, \label{e.6.19} \end{equation}
  </div>
  <span class="equation_label">5.71</span>
</p>
</div>
<p>and</p>
<div class="equation" id="e.6.20">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle A^{-2r}x,x\right\rangle \leq \left\langle A^{-r-1}x,x\right\rangle ^{r}\left\langle A^{-r}x,x\right\rangle ^{1-r}, \label{e.6.20} \end{equation}
  </div>
  <span class="equation_label">5.72</span>
</p>
</div>
<p>for any \(x\in H,\) \(\left\Vert x\right\Vert =1,\) where \(A\) is a positive definite operator on \(H\) and \(r\in \left( 0,1\right) .\) <small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
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</a> </p>
</dd>
  <dt><a name="Al">2</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1090/s0002-9939-2011-11088-8"> <i class="sc">J. M. Aldaz</i>, <i class="itshape">A measure-theoretic version of the Dragomir-Jensen inequality.</i> Proc. Amer. Math. Soc. <b class="bfseries">140</b> (2012), no. 7, 2391–2399.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><a href ="http://dx.doi.org/10.15352/bjma/1358864553"> <i class="sc">V. Cirtoaje</i>, <i class="itshape">The best lower bound for Jensen’s inequality with three fixed ordered variables.</i> Banach J. Math. Anal. <b class="bfseries">7</b> (2013), no. 1, 116–131.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="D0">4</a></dt>
  <dd><p><i class="sc">S. S. Dragomir</i>, <i class="itshape">A converse result for Jensen’s discrete inequality via Gruss’ inequality and applications in information theory.</i> An. Univ. Oradea Fasc. Mat. <b class="bfseries">7</b> (1999/2000), 178–189. </p>
</dd>
  <dt><a name="D">5</a></dt>
  <dd><p><i class="sc">S. S. Dragomir</i>, <i class="itshape">On a reverse of Jessen’s inequality for isotonic linear functionals,</i> J. Ineq. Pure &amp; Appl. Math., <b class="bfseries">2 </b>(2001), No. 3, Article 36. </p>
</dd>
  <dt><a name="D1">6</a></dt>
  <dd><p><a href ="http://rgmia.org/v5(E).php"> <i class="sc">S. S. Dragomir</i>, <i class="itshape">A Grüss type inequality for isotonic linear functionals and applications.</i> Demonstratio Math. <b class="bfseries">36</b> (2003), no. 3, 551–562. Preprint RGMIA Res. Rep. Coll. <b class="bfseries">5 </b>(2002), Suplement, Art. 12. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="SSD">7</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1017/s0004972708000671"> <i class="sc">S. S. Dragomir</i>, <i class="itshape">Bounds for the deviation of a function from the chord generated by its extremities.</i> Bull. Aust. Math. Soc. <b class="bfseries">78</b> (2008), no. 2, 225–248.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="SSDBul">8</a></dt>
  <dd><p><i class="sc">S. S. Dragomir</i>, <i class="itshape">Bounds for the normalized Jensen functional.</i> Bull. Austral. Math. Soc. <b class="bfseries">74 </b>(3) (2006), 471-476. </p>
</dd>
  <dt><a name="SSD1">9</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s40306-013-0029-9"> <i class="sc">S. S. Dragomir</i>, <i class="itshape">Reverses of the Jensen inequality in terms of first derivative and applications.</i> Acta Math. Vietnam. <b class="bfseries">38</b> (2013), no. 3, 429–446.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="SSD2">10</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1017/s0004972712001098"> <i class="sc">S. S. Dragomir</i>, <i class="itshape">Some reverses of the Jensen inequality with applications.</i> Bull. Aust. Math. Soc. <b class="bfseries">87</b> (2013), no. 2, 177–194.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="SSD3">11</a></dt>
  <dd><p><a href ="http://rgmia.org/papers/v14/v14a74.pdf"> <i class="sc">S. S. Dragomir</i>, <i class="itshape">A refinement and a divided difference reverse of Jensen’s inequality with applications</i>, accepted Rev. Colomb. Mate., Preprint RGMIA Res. Rep Coll. <b class="bfseries">14</b> (2011), Art 74.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="DI">12</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art7"> <i class="sc">S. S. Dragomir</i> and <i class="sc">N. M. Ionescu</i>, <i class="itshape">Some converse of Jensen’s inequality and applications.</i> Rev. Anal. Numér. Théor. Approx. <b class="bfseries">23</b> (1994), no. 1, 71–78.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="H">13</a></dt>
  <dd><p><i class="sc">G. Helmberg</i>,<i class="itshape"> Introduction to Spectral Theory in Hilbert Space,</i> John Wiley, New York, 1969. </p>
</dd>
  <dt><a name="Ho">14</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1186/1029-242x-2013-551"> <i class="sc">L. Horváth</i>, <i class="itshape">A new refinement of discrete Jensen’s inequality depending on parameters.</i> J. Inequal. Appl. <b class="bfseries">2013</b>, 2013:551, 16 pp.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="KKP">15</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.5666/kmj.2012.52.4.495"> <i class="sc">K. M. Adil, S. Khalid</i> and <i class="sc">J. Pečarić</i>, <i class="itshape">Improvement of Jensen’s inequality in terms of Gâteaux derivatives for convex functions in linear spaces with applications</i>. Kyungpook Math. J. <b class="bfseries">52</b> (2012), no. 4, 495–511.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Mi">16</a></dt>
  <dd><p><i class="sc">M. V. Mihai</i>, <i class="itshape">Jensen’s inequality for fixed convex functions of two real variables</i>. Acta Univ. Apulensis Math. Inform. No. <b class="bfseries">34</b> (2013), 179–183. </p>
</dd>
  <dt><a name="M">17</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.laa.2012.12.011"> <i class="sc">M. Kian</i>, <i class="itshape">Operator Jensen inequality for superquadratic functions</i>. Linear Algebra Appl. <b class="bfseries">456</b> (2014), 82–87.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="N">18</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1155/s1025583401000273"> <i class="sc">C. P. Niculescu</i>, <i class="itshape">An extension of Chebyshev’s inequality and its connection with Jensen’s inequality.</i> J. Inequal. Appl. <b class="bfseries">6</b> (2001), no. 4, 451–462.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="P">19</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1155/2013/194830"> <i class="sc">Z. Pavić</i>, <i class="itshape">The applications of functional variants of Jensen’s inequality.</i> J. Funct. Spaces Appl. <b class="bfseries">2013</b>, Art. ID 194830, 5 pp.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="PS">20</a></dt>
  <dd><p><i class="sc">F. Popovici</i> and <i class="sc">C.-I. Spiridon</i>, <i class="itshape">The Jensen inequality for </i>\((M,N)\)<i class="itshape">-convex functions.</i> An. Univ. Craiova Ser. Mat. Inform. <b class="bfseries">38</b> (2011), no. 4, 63–66. </p>
</dd>
  <dt><a name="Sh">21</a></dt>
  <dd><p><i class="sc">R. Sharma</i>, <i class="itshape">On Jensen’s inequality for positive linear functionals.</i> Int. J. Math. Sci. Eng. Appl. <b class="bfseries">5</b> (2011), no. 5, 263–271. </p>
</dd>
  <dt><a name="SS">22</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.jmaa.2008.01.060"> <i class="sc">S. Simić</i>, <i class="itshape">On a global upper bound for Jensen’s inequality</i>, J. Math. Anal. Appl. <b class="bfseries">343</b>(2008), 414-419.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="ZK">23</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1186/1029-242x-2012-238"> <i class="sc">G. Zabandan</i> and <i class="sc">A. Kılıçman</i>, <i class="itshape">A new version of Jensen’s inequality and related results.</i> J. Inequal. Appl. <b class="bfseries">2012</b>, 2012:238, 7 pp.<img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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