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<title>On the refinements of Jensen-Mercer’s inequality\(^\bullet \): On the refinements of Jensen-Mercer’s inequality\(^\bullet \)</title>
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<h1>On the refinements of Jensen-Mercer’s inequality\(^\bullet \)</h1>
<p class="authors">
<span class="author">M. Adil Khan,\(^{\ltimes ,\triangledown }\) Asif R. Khan\(^{\ast ,\triangledown }\) J. Pečarić\(^{\S ,\triangledown }\)</span>
</p>
<p class="date">January 30, 2012.</p>
</div>
<p>\(^\ltimes \)Department of Mathematics, University of University of Peshawar, Pakistan,<br />e-mail: <span class="tt">adilbandai@yahoo.com</span>. </p>
<p>\(^\triangledown \) Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan </p>
<p>\(^\ast \)Department of Mathematical Sciences, University of Karachi, University Road, Karachi, Pakistan, e-mail: <span class="tt">asif_rizkhan@yahoo.com</span>. </p>
<p>\(^\S \)University of Zagreb, Faculty of Textile Technology Zagreb, Croatia,<br />e-mail: <span class="tt">pecaric@mahazu.hazu.hr</span>. </p>
<p>\(^\bullet \)This research work is funded by Higher Education Commission Pakistan. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888. </p>

<div class="abstract"><p> In this paper we give refinements of Jensen-Mercer’s inequality and its generalizations and give applications for means. We prove \(n\)-exponential convexity of the functions constructed from these refinements. At the end we discuss some examples. </p>
<p><b class="bf">MSC.</b> 26D15 </p>
<p><b class="bf">Keywords.</b> convex functions, Jensen’s Mercer’s inequality, \(n\)-exponential convexity. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>In paper <span class="cite">
	[
	<a href="#mercer" >8</a>
	]
</span> A. McD. Mercer proved the following variant of Jensen’s inequality, to which we will refer as to the Jensen-Mercer inequality. <div class="theorem_thmwrapper " id="thMercerJenine">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \([a,b]\) be an interval in \(\mathbb {R}\), and \(x_1,...,x_n\in [a,b]\). Let \(w_1,w_2,...,w_n\) be nonnegative real numbers such that \(\sum _{i=1}^nw_i=1\). If \(\phi \) is a convex function on \([a,b]\), then </p>
<div class="equation" id="MercerJenine">
<p>
  <div class="equation_content">
    \begin{equation} \label{MercerJenine} \phi \left(a+b-\sum _{i=1}^nw_ix_i\right)\leq \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i). \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>

  </div>
</div> Given two real row \(n\)-tuples \(\mathbf{x}=(x_1,...,x_n)\) and \(\mathbf{y}=(y_1,...,y_n)\), \(\mathbf{y}\) is said to majorize \(\mathbf{x}\), if </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  \sum _{i=1}^k x_{[i]}\, \leq \, \sum _{i=1}^k y_{[i]} \end{equation*}
</div>
<p> holds for \( k =1, 2, ..., n-1\) and </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \sum _{i=1}^n x_{i}\, =\, \sum _{i=1}^n y_{i}, \end{equation*}
</div>
<p> where \( x_{[1]}\geq ...\geq \,  x_{[n]},\, \mbox{ and } y_{[1]}\geq ...\geq \,  y_{[n]},\) are the entries of \(\mathbf{x}\) and \(\mathbf{y}\), respectively, in nonincreasing order (see <span class="cite">
	[
	<a href="#marshal" >6</a>
	, 
	p. 10
	]
</span>). <br /></p>
<p>The following extension of (<a href="#MercerJenine">1</a>) is given in <span class="cite">
	[
	<a href="#niezgodamercer" >9</a>
	]
</span>. <div class="theorem_thmwrapper " id="a0000000005">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a continuous convex function on \([a,b]\). Suppose that \(\textbf{a}=(a_1,...,a_m)\) with \(a_j\in [a,b]\), and \(\textbf{X}=(x_{ij})\) is a real \(n\times m\) matrix such that \(x_{ij}\in [a,b]\) for all \(i=1,\ldots ,n;\, \, j=1,\ldots ,m\). </p>
<p>If \(\textbf{a}\) majorizes each row of \(\textbf{X}\), that is </p>
<div class="displaymath" id="a0000000006">
  \[ \textbf{x}_{i.}=(x_{i1},...,x_{im})\prec (a_1,...,a_m)=\textbf{a}\mbox{ for each }i=1,...,n; \]
</div>
<p> then we have the inequality </p>
<div class="equation" id="niezgodaineq">
<p>
  <div class="equation_content">
    \begin{equation} \label{niezgodaineq} \phi \left(\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\right)\leq \sum _{j=1}^{m}\phi (a_j)-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\phi (x_{ij}), \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \(\sum _{i=1}^{n}w_i=1\) with \(w_{i}\geq 0\). </p>

  </div>
</div> In this paper we give refinements of (<a href="#MercerJenine">1</a>), (<a href="#niezgodaineq">2</a>) and give applications for means. We construct functionals from these refinements and prove mean value theorems. The notion of \(n\)-exponential convexity is introduced in <span class="cite">
	[
	<a href="#nexp" >10</a>
	]
</span>. The class of \(n\)-exponential convex functions is more general than the class of log-convex functions. We follow the method illustrated in <span class="cite">
	[
	<a href="#nexp" >10</a>
	]
</span> to give the \(n\)-exponential convexity and exponential convexity for these functionals. </p>
<h1 id="a0000000007">2 Main results</h1>
<p>Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a convex function. If \(x_i\in [a,b]\) and \(w_i{\gt}0\), \(i\in \{ 1,2,...,n\} \) with \(\sum _{i=1}^nw_i=1\). Throughout the paper we assume that \(I\subset \{ 1,2,...,n\} \) with \(I\neq \emptyset \) and \(I\neq \{ 1,2,...,n\} \) unless stated. We define \(W_I=\sum _{i \in I}w_i\) and \(W_{\overline{I}}=1-\sum _{i \in {I}}w_i\). For the convex function \(\phi \) and the \(n\)-tuple \(\mathbf{x}=(x_1,...,x_n)\) and \(\mathbf{w}=(w_1,...,w_n)\) as above, we can define the following functional </p>
<div class="equation" id="spefunctional">
<p>
  <div class="equation_content">
    \begin{equation} \label{functional} D(\mathbf{w},\mathbf{x},\phi ;I):=W_I\phi \left(a+b-\tfrac {1}{W_I}\sum _{i \in I}w_ix_i\right)+W_{\overline{I}}\phi \left(a+b-\tfrac {1}{W_{\overline{I}}}\sum _{i \in \overline{I}}w_ix_i\right). \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> It is worth to observe that for \(I=\{ k\} \), \(k\in \{ 1,...,n\} \) we have the functional </p>
<div class="displaymath" id="a0000000008">
  \begin{align*} \label{spefunctional} D_k(\mathbf{w},\mathbf{x},\phi ) & :=D(\mathbf{w},\mathbf{x},\phi ;\{ k\} )\nonumber \\ & =w_k\phi (a+b-x_k)+(1-w_k)\phi \left(a+b-\tfrac {\sum _{i=1}^nw_ix_i-w_kx_k}{1-w_k}\right). \end{align*}
</div>
<p>The following refinement of (<a href="#MercerJenine">1</a>) is valid. <div class="theorem_thmwrapper " id="thMercerJenineref">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \([a,b]\) be an interval in \(\mathbb {R}\), and \(x_1,...,x_n\in [a,b]\). Let \(w_1,w_2,...,w_n\) be positive real numbers such that \(\sum _{i=1}^nw_i=1\). If \(\phi :[a,b]\rightarrow \mathbb {R}\) is a convex function, then for any non empty subset \(I\) of \(\{ 1,...,n\} \) we have </p>
<div class="equation" id="MercerJenineref">
<p>
  <div class="equation_content">
    \begin{equation} \label{MercerJenineref} \phi \left(a+b-\sum _{i=1}^nw_ix_i\right)\leq D(\mathbf{w},\mathbf{x},\phi ;I)\leq \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i). \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By the convexity of the function \(\phi \) we have </p>
<div class="displaymath" id="a0000000010">
  \begin{align*} & \phi \Big(a+b-\sum _{i=1}^nw_ix_i\Big) = \phi \Big(\sum _{i=1}^nw_i\Big(a+b-x_i\Big)\Big) \\ & = \phi \Big(W_I\Big(\tfrac {1}{W_I}\sum _{i \in I}w_i\Big(a+b-x_i\Big)\Big)+W_{\overline{I}}\Big(\tfrac {1}{W_{\overline{I}}}\sum _{i \in \overline{I}}w_i\Big(a+b-x_i\Big)\Big)\Big)\\ & \leq W_I\phi \Big(\tfrac {1}{W_I}\sum _{i \in I}w_i\Big(a+b-x_i\Big)\Big)+W_{\overline{I}}\phi \Big(\tfrac {1}{W_{\overline{I}}}\sum _{i \in \overline{I}}w_i\Big(a+b-x_i\Big)\Big)\\ & =D(\mathbf{w},\mathbf{x},\phi ;I)\hspace{8.7cm} \end{align*}
</div>
<p> for any \(I\), which proves the first inequality in (<a href="#MercerJenineref">3</a>). </p>
<p>By the Jensen-Mercer inequality (<a href="#MercerJenine">1</a>) we also have </p>
<div class="displaymath" id="a0000000011">
  \begin{align*} & D(\mathbf{w},\mathbf{x},\phi ;I)=W_{I}\phi \Big(a+b-\tfrac {1}{W_{I}}\sum _{i\in I}w_ix_i\Big)+ W_{\overline{I}}\phi \Big(a+b-\tfrac {1}{W_{\overline{I}}}\sum _{i\in \overline{I}}w_ix_i\Big)\\ & \leq W_{I}\Big(\phi (a)+\phi (b)-\tfrac {1}{W_{I}}\sum _{i\in I}w_i\phi (x_i)\Big)+ W_{\overline{I}}\Big(\phi (a)+\phi (b)-\tfrac {1}{W_{\overline{I}}}\sum _{i\in \overline{I}}w_i\phi (x_i)\Big) \\ & = \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i)\hspace{-4cm}. \end{align*}
</div>
<p> for any \(I\), which proves the second inequality in (<a href="#MercerJenineref">3</a>). <div class="proof_wrapper" id="a0000000012">
  <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="rk_thmwrapper " id="a0000000013">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">4</span>
  </div>
  <div class="rk_thmcontent">
  <p>In <span class="cite">
	[
	<a href="#pecaricref" >7</a>
	]
</span> from the proof of Theorem 2.3 we have left inequality of (<a href="#MercerJenineref">3</a>).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="rk_thmwrapper " id="rk1">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">5</span>
  </div>
  <div class="rk_thmcontent">
  <p> We observe that the inequality (<a href="#MercerJenineref">3</a>) can be written in an equivalent form as </p>
<div class="equation" id="min">
<p>
  <div class="equation_content">
    \begin{equation} \label{min} \phi \left(a+b-\sum _{i=1}^nw_ix_i\right)\leq \min \limits _{I} D(\mathbf{w},\mathbf{x},\phi ;I) \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> and </p>
<div class="equation" id="spefuncunif">
<p>
  <div class="equation_content">
    \begin{equation} \label{max} \max \limits _{I} D(\mathbf{w},\mathbf{x},\phi ;I)\leq \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i). \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> The following special cases of (<a href="#min">4</a>) and (<a href="#spefuncunif">5</a>) can be given: </p>
<div class="displaymath" id="a0000000014">
  \begin{equation*} \label{minspe} \phi \left(a+b-\sum _{i=1}^nw_ix_i\right)\leq \min \limits _{ k\in \{ 1,...,n\} } D_k(\mathbf{w},\mathbf{x},\phi ) \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000015">
  \begin{equation*} \label{maxspe} \max \limits _{k\in \{ 1,...,n\} } D_k(\mathbf{w},\mathbf{x},\phi )\leq \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i).\hfil \qed \end{equation*}
</div>

  </div>
</div> </p>
<p>The case of uniform distribution, namely, when \(w_i=\tfrac {1}{n}\) for all \(i=1,2,...,n\) is of interest as well. If we consider a natural number \(m\) with \( m\in \{ 1,2,\ldots ,n-1\} \) and if we define </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*} \label{spefuncunif} D_m(\mathbf{x},\phi ):=\tfrac {m}{n}\phi \left(a+b-\tfrac {1}{m}\sum _{i=1}^mx_i\right)+\tfrac {n-m}{n}\phi \left(a+b-\tfrac {1}{n-m}\sum _{j=m+1}^nx_j\right) \end{equation*}
</div>
<p> then we can state the following result: <div class="cor_thmwrapper " id="Mercerrefspecialcas">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">6</span>
  </div>
  <div class="cor_thmcontent">
  <p>If \(\phi :[a,b]\rightarrow \mathbb {R}\) is a convex function, \(x_i\in [a,b]\), \(i\in \{ 1,2,...,n\} \), then for any \(m\in \{ 1,2,...,n-1\} \) we have </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*} \label{Mercerrefspecialcas} \phi \left(a+b-\tfrac {1}{n}\sum _{i=1}^nx_i\right)\leq D_m(\mathbf{x},\phi )\leq \phi (a)+\phi (b)-\tfrac {1}{n}\sum _{i=1}^n\phi (x_i). \end{equation*}
</div>
<p> In particular, we have the bounds </p>
<div class="displaymath" id="a0000000018">
  \[  \phi \left(a+b-\tfrac {1}{n}\sum _{i=1}^nx_i\right) \leq \min \limits _{m\in \{ 1,...,n-1\} }D_m(\mathbf{x},\phi )  \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000019">
  \[  \max \limits _{m\in \{ 1,...,n-1\} }D_m(\mathbf{x},\phi )\leq \phi (a)+\phi (b)-\tfrac {1}{n}\sum _{i=1}^n\phi (x_i).  \]
</div>

  </div>
</div> </p>
<p>The following refinement of (<a href="#niezgodaineq">2</a>) is valid. <div class="theorem_thmwrapper " id="theniezref">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a continuous convex function on \([a,b]\). Suppose that \(\textbf{a}=(a_1,...,a_m)\) with \(a_j\in [a,b]\), and \(\textbf{X}=(x_{ij})\) is a real \(n\times m\) matrix such that \(x_{ij}\in [a,b]\) for all \(i=1,\ldots ,n;\, \, j=1,\ldots ,m\). </p>
<p>If \(\textbf{a}\) majorizes each row of \(\textbf{X}\), then for any non empty subset \(I\) of \(\{ 1,...,n\} \) we have </p>
<div class="equation" id="niezgodaineqref">
<p>
  <div class="equation_content">
    \begin{equation} \label{niezgodaineqref} \phi \left(\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\right)\leq \tilde{D}(\mathbf{w},\mathbf{X},\phi ;I)\leq \sum _{j=1}^{m}\phi (a_j)-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\phi (x_{ij}), \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="tilD">
  \begin{align} \label{tilD} & \tilde{D}(\mathbf{w},\mathbf{X},\phi ;I):=\\ & =W_{I} \phi \left(\sum _{j=1}^{m}a_j-\tfrac {1}{W_I}\sum _{j=1}^{m-1}\sum _{i\in I}w_ix_{ij}\right)+ W_{\overline{I}} \phi \left( \sum _{j=1}^{m}a_j- \tfrac {1}{W_{\overline{I}}}\sum _{j=1}^{m-1} \sum _{i\in \overline{I}}w_ix_{ij}\right),\nonumber \end{align}
</div>
<p> \(W_{I}=\sum _{i\in I}w_i, W_{\overline{I}}=\sum _{i\in \overline{I}}w_i\), \(\sum _{i=1}^nw_i=1\) with \(w_{i}{\gt}0\). </p>

  </div>
</div> </p>
<p><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> The proof is similar to the proof of Theorem <a href="#thMercerJenineref">3</a> but use (<a href="#niezgodaineq">2</a>) instead of (<a href="#MercerJenine">1</a>). <div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> As above we can give the following remark. <div class="rk_thmwrapper " id="a0000000022">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">8</span>
  </div>
  <div class="rk_thmcontent">
  <div class="displaymath" id="a0000000023">
  \[ \phi \left(\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\right)\leq \min \limits _{I}\tilde{D}(\mathbf{w},\mathbf{X},\phi ;I) \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000024">
  \[  \max \limits _{I}\tilde{D}(\mathbf{w},\mathbf{X},\phi ;I)\leq \sum _{j=1}^{m}\phi (a_j)-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\phi (x_{ij}).\hfil \qed  \]
</div>

  </div>
</div> </p>
<p><div class="rk_thmwrapper " id="a0000000025">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">9</span>
  </div>
  <div class="rk_thmcontent">
  <p>If in (<a href="#niezgodaineq">2</a>) we set \(m=2, a_1=a, a_2=b\) and \(x_{i1}=x_i\) for \(i=1,...,n\) we get (<a href="#MercerJenineref">3</a>).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>An \(m \times m\) matrix \(\textbf{A}=(a_{jk})\) is said to be doubly stochastic, if \(a_{jk} \geq 0\) and \(\sum _{j=1}^{m}a_{jk}=\sum _{k=1}^{m}a_{jk}=1\) for all \(j,k=1,...,m\). It is well known <span class="cite">
	[
	<a href="#marshal" >6</a>
	, 
	p. 20
	]
</span> that if \(\textbf{A}\) is an \(m\times m\) doubly stochastic matrix, then </p>
<div class="equation" id="stocheq">
<p>
  <div class="equation_content">
    \begin{equation} \label{stocheq} \textbf{a}\textbf{A}\prec \textbf{a}\mbox{ for each real $m$-tuple } \textbf{a}=(a_1,a_2,...,a_m). \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>By applying Theorem <a href="#theniezref">7</a> and (<a href="#stocheq">8</a>), one obtains: <div class="cor_thmwrapper " id="a0000000026">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">10</span>
  </div>
  <div class="cor_thmcontent">
  <p>Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a continuous convex function on \([a,b]\). Suppose that \(\textbf{a}=(a_1,...,a_m)\) with \(a_j\in [a,b]\) \(j=1,...,m\) and \(\textbf{A}_1, \textbf{A}_2,...,\textbf{A}_n\) are \(m\times m\) doubly stochastic matrices. Set </p>
<div class="displaymath" id="a0000000027">
  \[ X=(x_{ij})=\begin{pmatrix}  \textbf{a}\textbf{A}_1 

\\ . 

\\ . 

\\ \textbf{a}\textbf{A}_n 

\\ \end{pmatrix}.  \]
</div>
<p> Then inequalities in <span class="rm">(<a href="#niezgodaineqref">6</a>)</span> hold. </p>

  </div>
</div> <div class="rk_thmwrapper " id="a0000000028">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">11</span>
  </div>
  <div class="rk_thmcontent">
  <p>In <span class="cite">
	[
	<a href="#dragnerefjen" >4</a>
	]
</span> Dragomir has given related refinements of Jensen’s inequality.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000029">3 Applications</h1>
<p> For \(\emptyset \neq I\subseteq \{ 1,...,n\} \) let \(A_I,G_I,H_I\) and \(M^{[r]}_I\) be the arithmetic, geometric, harmonic means, and power mean of order \(r\in \mbox{\black R}\), respectively of \(x_i\in [a,b]\), where \(0 {\lt} a {\lt} b\), formed with the positive weights \(w_i\), \(i\in I\). For \(I=\{ 1,...,n\} \) we denote the arithmetic, geometric, harmonic and power means by \(A_n,G_n,H_n\) and \(M^{[r]}_n\) respectively. </p>
<p>If we define </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  \tilde{A}_I:& =& a+b-\tfrac {1}{W_{I}}\sum _{i\in I}w_ix_i=a+b-A_I\\ \tilde{G}_I:& =& \tfrac {ab}{\left(\prod \limits _{i\in I} x_i^{w_i}\right)^{\tfrac {1}{W_I}}}=\tfrac {ab}{G_I}\\ \tilde{H}_I:& =& \left(a^{-1}+b^{-1}-\tfrac {1}{W_{I}}\sum _{i\in I}w_ix^{-1}_i\right)^{-1}=\left(a^{-1}+b^{-1}-H^{-1}_I\right)^{-1}\\ \tilde{M}^{[r]}_I:& =& \begin{cases} \left(a^r+b^r-\left(M^{[r]}_I\right)^{r}\right)^{\tfrac {1}{r}}, & r\neq 0;\\ \tilde{G}_I, & r=0, \end{cases}\end{eqnarray*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000031">
  \[  {M}^{[r]}_I:=\begin{cases} \left(\tfrac {1}{W_{I}}\sum _{i\in I}w_ix^{r}_i\right)^{\tfrac {1}{r}}, & r\neq 0;\\ \left(\prod \limits _{i\in I} x_i^{w_i}\right)^{\tfrac {1}{W_I}}, & r=0, \end{cases}  \]
</div>
<p> then the following inequalities hold. </p>
<p><div class="theorem_thmwrapper " id="theappl">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  
<div class="displaymath" id="app1">
  \begin{eqnarray} \label{app1} \quad (i) \, \, \, \, \tilde{G}_n \leq \min \limits _{I} \tilde{A}_I^{W_I} \tilde{A}_{\overline{I}}^{W_{\overline{I}}}\, \, \, \, \mbox{ and } \tilde{A}_n \geq \max \limits _{I} \tilde{A}_I^{W_I} \tilde{A}_{\overline{I}}^{W_{\overline{I}}}.\hspace{3cm} \end{eqnarray}
</div>
<div class="displaymath" id="app2">
  \begin{eqnarray} \label{app2} \quad (ii) \, \, \, \, \tilde{G}_n \leq \min \limits _{I}\left[W_I\tilde{G}_I+ W_{\overline{I}}\tilde{G}_{\overline{I}}\right]\,  \mbox{ and } \tilde{A}_n \geq \max \limits _{I}\left[W_I\tilde{G}_I+ W_{\overline{I}}\tilde{G}_{\overline{I}}\right]. \end{eqnarray}
</div>

  </div>
</div> <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> (i) Applying Theorem <a href="#thMercerJenineref">3</a> to the convex function \(\phi (x)=-\ln x\), we obtain </p>
<div class="equation" id="sol1">
<p>
  <div class="equation_content">
    \begin{equation} \label{sol1} -\ln \tilde{A}_n\leq - W_I\ln \tilde{A}_I- W_{\overline{I}}\ln \tilde{A}_{\overline{I}} \leq - \ln \tilde{G}_n. \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> Now (<a href="#app1">13</a>) follows from Remark <a href="#rk1">5</a> and (<a href="#sol1">15</a>).<br /><br />(ii) Applying Theorem <a href="#thMercerJenineref">3</a> to the convex function \(\phi (x)=\exp (x)\), and replacing \(a, b\), and \(x_i\) with \(\ln a, \ln b\), and \(\ln x_i\) respectively and using Remark <a href="#rk1">5</a>, we obtain (<a href="#app2">14</a>). <div class="proof_wrapper" id="a0000000033">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The following particular case of Theorem <a href="#theappl">12</a> is of interest. <div class="cor_thmwrapper " id="corapp2">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">13</span>
  </div>
  <div class="cor_thmcontent">
  <div class="displaymath" id="a0000000034">
  \begin{eqnarray*}  (i) \, \, \, \, \tfrac {1}{\tilde{G_n}}\leq \min \limits _{I} \tfrac {1}{\tilde{H}^{W_I}_I\tilde{H_{\overline{I}}}^{W_{\overline{I}}}}\, \, \, \, \mbox{ and }\, \, \, \,  \tfrac {1}{\tilde{H}_n}\geq \max \limits _{I} \tfrac {1}{\tilde{H}^{W_I}_I\tilde{H_{\overline{I}}}^{W_{\overline{I}}}}.\hspace{1cm} \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000035">
  \begin{eqnarray*} \label{corapp2} (ii) \, \, \, \, \tfrac {1}{\tilde{G}_n}\leq \min \limits _{I} \left[ \tfrac {W_I}{\tilde{G}_I}+ \tfrac {W_{\overline{I}}}{\tilde{G}_{\overline{I}}}\right]\, \, \, \, \mbox{ and }\, \, \, \,  \tfrac {1}{\tilde{H}_n}\geq \max \limits _{I}\left[\tfrac {W_I}{\tilde{G}_I}+ \tfrac {W_{\overline{I}}}{\tilde{G}_{\overline{I}}}\right]. \end{eqnarray*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000036">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Directly from Theorem <a href="#theappl">12</a> by the substitutions \(a\rightarrow \tfrac {1}{a}, b\rightarrow \tfrac {1}{b},x_i\rightarrow \tfrac {1}{x_i}\). <div class="proof_wrapper" id="a0000000037">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="theorem_thmwrapper " id="thpmeanqua">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For \(r\leq 1\), we have the following inequalities </p>
<div class="displaymath" id="pmean">
  \begin{eqnarray} \label{pmean} \tilde{M}^{[r]}_n\leq \min \limits _{I} \left[W_I \tilde{M}^{[r]}_I +W_{\overline{I}}\tilde{M}^{[r]}_{\overline{I}}\right],\nonumber \\ \tilde{A}_n\geq \max \limits _{I} \left[W_I \tilde{M}^{[r]}_I +W_{\overline{I}}\tilde{M}^{[r]}_{\overline{I}}\right]. \end{eqnarray}
</div>
<p> For \(r\geq 1\), the inequalities in <span class="rm">(<a href="#pmean">16</a>)</span> are reversed. </p>

  </div>
</div> <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> For \(r\leq 1\), \(r\neq 0\), use Theorem <a href="#thMercerJenineref">3</a> for the convex function \(\phi (x)=x^{\tfrac {1}{r}}\), and replacing \(a,b,\) and \(x_i\) with \(a^r,b^r,\) and \(x_i^r\) respectively and for \(r=0\) use Theorem <a href="#thMercerJenineref">3</a> for the convex function \(\phi (x)=\exp (x)\); replacing \(a,b,\) and \(x_i\) with \(\ln a,\ln b,\) and \(\ln x_i\) respectively, we obtain (<a href="#pmean">16</a>) by Remark <a href="#rk1">5</a>.<br />If \(r\geq 1\), then the function \(\phi (x)=x^{\tfrac {1}{r}}\) is concave, so the inequalities in (<a href="#pmean">16</a>) are reversed. <div class="proof_wrapper" id="a0000000039">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="cor_thmwrapper " id="a0000000040">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">15</span>
  </div>
  <div class="cor_thmcontent">
  <div class="displaymath" id="a0000000041">
  \begin{eqnarray*}  \tilde{H}_n\leq \min \limits _{I} \left[W_I \tilde{H}_I +W_{\overline{I}}\tilde{H}_{\overline{I}}\right],\nonumber \\ \tilde{A}_n\geq \max \limits _{I} \left[W_I \tilde{H}_I +W_{\overline{I}}\tilde{H}_{\overline{I}}\right]. \end{eqnarray*}
</div>

  </div>
</div> <div class="rk_thmwrapper " id="a0000000042">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">16</span>
  </div>
  <div class="rk_thmcontent">
  <p>Obviously, part (ii) of Theorem <a href="#theappl">12</a> is also a direct consequences of Theorem <a href="#thpmeanqua">14</a>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="thpmean">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">17</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(r,s\in \mathbb {R}\), \(r\leq s\). </p>
<p>(i) If \(s\geq 0,\) then </p>
<div class="displaymath" id="pmean2">
  \begin{eqnarray} \label{pmean2} \left(\tilde{M}^{[r]}_n\right)^s\leq \min \limits _{I} \left[W_I \left(\tilde{M}^{[r]}_I \right)^s +W_{\overline{I}} \left(\tilde{M}^{[r]}_{\overline{I}}\right)^s\right],\nonumber \\ \left(\tilde{M}^{[r]}_n\right)^s\geq \max \limits _{I} \left[W_I \left(\tilde{M}^{[r]}_I \right)^s +W_{\overline{I}} \left(\tilde{M}^{[r]}_{\overline{I}}\right)^s\right]. \end{eqnarray}
</div>
<p> (ii) If \(s{\lt}0\), then inequalities in <span class="rm">(<a href="#pmean2">16</a>)</span> are reversed. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000043">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(s\geq 0\). Using Theorem <a href="#thMercerJenineref">3</a> and Remark <a href="#rk1">5</a> to the convex function \(\phi (x)=x^{\tfrac {s}{r}},\) and replacing \(a,b,\) and \(x_i\) with \(a^r,b^r,\) and \(x_i^r\) respectively, we obtain (<a href="#pmean2">16</a>).<br />If \(s{\lt}0\), then the function \(\phi (x)=x^{\tfrac {s}{r}},\) is concave so inequalities in (<a href="#pmean2">16</a>) are reversed. <div class="proof_wrapper" id="a0000000044">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a strictly monotonic and continuous function. Then for a given \(n\)- tuple \(\textbf{x}=(x_1,...,x_n)\in [a,b]^n\) and positive \(n\)- tuple \(\textbf{w}=(w_1,...,w_n)\) with \(\sum _{i=1}^{n} w_i=1\), the value </p>
<div class="displaymath" id="a0000000045">
  \[ M^{[n]}_{\phi }=\phi ^{-1}\left(\sum _{i=1}^{n}w_i\phi (x_i)\right) \]
</div>
<p> is well defined and is called \(quasi-arithmetic\  mean\) of \(\textbf{x}\) with wight \(\textbf{w}\) (see for example <span class="cite">
	[
	<a href="#Mitrinovic" >2</a>
	, 
	p. 215
	]
</span>). If we define </p>
<div class="displaymath" id="a0000000046">
  \[ \tilde{M}^{[n]}_{\phi }=\phi ^{-1}\left(\phi (a)+\phi (b)-\sum _{i=1}^{n}w_i\phi (x_i)\right), \]
</div>
<p> then we have the following results. <div class="theorem_thmwrapper " id="thqmean">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">18</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\phi ,\psi :[a,b]\rightarrow \mathbb {R}\) be strictly monotonic and continuous functions. If \(\psi \circ \phi ^{-1}\) is convex on \([a,b]\), then </p>
<div class="displaymath" id="qmean">
  \begin{eqnarray} \label{qmean} \psi \left(\tilde{M}^{[n]}_{\phi }\right)\leq \min \limits _{I} \left[W_I \psi \left(\tilde{M}^{[I]}_{\phi } \right) +W_{\overline{I}} \psi \left(\tilde{M}^{[\overline{I}]}_{\phi }\right)\right],\nonumber \\ \psi \left(\tilde{M}^{[n]}_{\phi }\right)\geq \max \limits _{I} \left[W_I \psi \left(\tilde{M}^{[I]}_{\phi } \right) +W_{\overline{I}} \psi \left(\tilde{M}^{[\overline{I}]}_{\phi }\right)\right], \end{eqnarray}
</div>
<div class="displaymath" id="a0000000047">
  \[ \mbox{where }\tilde{M}^{[J]}_{\phi }=\phi ^{-1}\left(\phi (a)+\phi (b)-\tfrac {1}{W_J}\sum _{i\in J}w_i\phi (x_i)\right). \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000048">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Applying Theorem <a href="#thMercerJenineref">3</a> to the convex function \(f=\psi \circ \phi ^{-1}\) and replacing \(a, b\), and \(x_i\) with \(\phi (a),\phi (b)\), and \(\phi (x_i)\) respectively and then using Remark <a href="#rk1">5</a>, we obtain (<a href="#qmean">17</a>). <div class="proof_wrapper" id="a0000000049">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="rk_thmwrapper " id="a0000000050">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">19</span>
  </div>
  <div class="rk_thmcontent">
  <p>Theorems <a href="#theappl">12</a>, <a href="#thpmeanqua">14</a> and <a href="#thpmean">17</a> follow from Theorem <a href="#thqmean">18</a>, by choosing adequate functions \(\phi \),\(\psi \) and appropriate substitutions.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000051">4 Further generalization</h1>
<p>Let \(E\) be a nonempty set, \(\mathfrak {A}\) be an algebra of subsets of \(E\), and \(L\) be a linear class of real valued functions \(f : E\rightarrow \mathbb {R}\) having the properties: </p>
<ul class="itemize">
  <li><p>: \(f,g \in L \Rightarrow (\alpha f+\beta g)\in L\) for all \(\alpha ,\beta \in \mathbb {R}\); </p>
</li>
  <li><p>: \(\mathbf{1}\in L\), i.e., if \(f(t)=1\) for all \(t\in E\), then \(f\in L\); </p>
</li>
  <li><p>: \(f\in L\), \(E_1\in \mathfrak {A} \Rightarrow f.\chi _{E_1}\in \mathfrak {A}\), </p>
</li>
</ul>
<p> where \(\chi _{E_1}\) is the indicator function of \(E_1\). It follows from \(L_2,L_3\) that \(\chi _{E_1}\in L\) for every \(E_1\in \mathfrak {A}\). <br /></p>
<p>An isotonic linear functional \(A:L\rightarrow \mathbb {R}\) is a functional satisfying the following properties: </p>
<ul class="itemize">
  <li><p>: \(A(\alpha f+\beta g)=\alpha A(f)+\beta A(g)\) for \(f,g\in L,\alpha ,\beta \in \mathbb {R}\); </p>
</li>
  <li><p>: \(f\in L, f(t)\geq 0\) on \(E\Rightarrow A(f)\geq 0\); </p>
</li>
</ul>
<p> It follows from \(L_3\) that for every \(E_1\in \mathfrak {A}\) such that \(A(\chi _{E_1}){\gt}0,\) the functional \(A_1\) defined for all \(f\in L\) as \(A_1(f)=\tfrac {A(f.\chi _{E_1})}{A(\chi _{E_1})}\) is an isotonic linear functional with \(A(\mathbf{1})=1\). Furthermore, we observe that </p>
<div class="displaymath" id="a0000000052">
  \begin{equation*} \label{a1} A(\chi _{E_1})+A(\chi _{E\setminus E_1})=1, \end{equation*}
</div>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  A(f)=A(f.\chi _{E_1})+A(f.\chi _{E\setminus E_1}).\\ \end{equation*}
</div>
<p> <br />Let \(\phi :[a,b]\rightarrow \mathbb {R}\) be a continuous function. In <span class="cite">
	[
	<a href="#Cheung" >3</a>
	]
</span>, under the above assumptions, the following variant of the Jessen inequality is proved, if \(\phi \) is convex, then </p>
<div class="equation" id="jensenmerfuntion">
<p>
  <div class="equation_content">
    \begin{equation} \label{jensenmerfuntion} \phi (a+b-A(f))\leq \phi (a)+\phi (b)-A(\phi (f)); \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> if \(\phi \) is concave then the inequality (<a href="#jensenmerfuntion">18</a>) is reversed. <br /></p>
<p>The following refinement of (<a href="#jensenmerfuntion">18</a>) holds. <div class="theorem_thmwrapper " id="threffunt">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">20</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the above assumptions, if \(\phi \) is convex, then </p>
<div class="displaymath" id="jensenmerreffunc">
  \begin{eqnarray} \label{jensenmerreffunc} \phi (a+b-A(f))\leq \overline{D}(A,f,\phi ;E_1)\leq \phi (a)+\phi (b)-A(\phi (f)); \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="overD">
  \begin{align} \label{overD} & \overline{D}(A,f,\phi ;E_1):=\\ =& A(\chi _{E_1})\phi \left(a+b-\tfrac {A(f.\chi _{E_1})}{A(\chi _{E_1})}\right)+ A(\chi _{E\setminus E_1})\phi \left(a+b-\tfrac {A(f.\chi _{E\setminus E_1})}{A(\chi _{E\setminus E_1})}\right)\nonumber \end{align}
</div>
<p> for all \(E_1\in \mathfrak {A}\) such that \(0{\lt}A(\chi _{E_1}){\lt}1\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000054">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The first inequality follows by using definition of convex function and the second follows by using (<a href="#jensenmerfuntion">18</a>) for \(A_1(f)\) instead of \(A(f)\). <div class="proof_wrapper" id="a0000000055">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="rk_thmwrapper " id="a0000000056">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">21</span>
  </div>
  <div class="rk_thmcontent">
  <p>In <span class="cite">
	[
	<a href="#pecaricref" >7</a>
	]
</span> from the proof of Theorem 4.1 we have left inequality of (<a href="#overD">20</a>).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="rk_thmwrapper " id="a0000000057">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">22</span>
  </div>
  <div class="rk_thmcontent">
  <p>We observe that the inequality (<a href="#jensenmerreffunc">19</a>) can be written in an equivalent form as </p>
<div class="displaymath" id="a0000000058">
  \begin{eqnarray*}  \phi (a+b-A(f)) \leq \min \limits _{ E_1\in \mathfrak {A}} \overline{D}(A,f,\phi ;E_1) \end{eqnarray*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*}  \phi (a)+\phi (b)-A(\phi (f)) \geq \max \limits _{E_1\in \mathfrak {A}} \overline{D}(A,f,\phi ;E_1).\hfil \qed \end{eqnarray*}
</div>

  </div>
</div> </p>
<p>The following particular case of Theorem <a href="#threffunt">20</a> is of interest: <div class="cor_thmwrapper " id="integappl">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">23</span>
  </div>
  <div class="cor_thmcontent">
  <p>Let \((\Omega , P, \mu )\) be a probability measure space, and let \(f:\Omega \rightarrow [a,b]\) be a measurable function. Then for any continuous convex function \(\phi : [a,b]\rightarrow \mathbb {R}\), and for any set \(E_1\) in \(P\) with \(\mu (E_1),\mu (\Omega \backslash E_1){\gt}0\) we have </p>
<div class="displaymath" id="a0000000060">
  \begin{align*} \label{integappl} \phi \left(a+b-\int _{\Omega }f{\rm d}\mu \right) & \leq \min \limits _{ E_1\in P} \left[ \mu (E_1)\phi \left(a+b-\tfrac {1}{\mu (E_1)}\int _{E_1}f{\rm d}\mu \right)\right.\nonumber \\ & \quad \left.+ \mu (\Omega \setminus E_1)\phi \left(a+b-\tfrac {1}{\mu (\Omega \setminus E_1)}\int _{\Omega \setminus E_1}f{\rm d}\mu \right)\right] \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000061">
  \begin{align*} \phi (a)+\phi (b)-\int _{\Omega }\phi (f){\rm d}\mu & \geq \max \limits _{ E_1\in P} \left[ \mu (E_1)\phi \left(a+b-\tfrac {1}{\mu (E_1)}\int _{E_1}f{\rm d}\mu \right)\right.\nonumber \\ & \quad \left.+ \mu (\Omega \setminus E_1)\phi \left(a+b-\tfrac {1}{\mu (\Omega \setminus E_1)}\int _{\Omega \setminus E_1}f{\rm d}\mu \right)\right]. \end{align*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000062">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> This is a special case of Theorem <a href="#threffunt">20</a> for the functional \(A\) defined on the class \(L^1(\mu )\) as \(A(f)=\int _{\Omega }f{\rm d}\mu \). <div class="proof_wrapper" id="a0000000063">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="rk_thmwrapper " id="a0000000064">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">24</span>
  </div>
  <div class="rk_thmcontent">
  <p>We also may obtain similar results as in Theorem <a href="#thqmean">18</a> for the generalized quasi-arithmetic means of Mercer’s type defined in <span class="cite">
	[
	<a href="#Cheung" >3</a>
	]
</span>, as </p>
<div class="displaymath" id="a0000000065">
  \[ \tilde{M}_{\phi }(f,A) = \phi ^{-1} (\phi (a) + \phi (b) - A(\phi (f))). \]
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000066">5 \(n\)-exponential convexity of the Jensen-Mercer differences</h1>
<p>Under the assumptions of Theorem <a href="#thMercerJenineref">3</a> using (<a href="#MercerJenineref">3</a>) we define the following functionals: </p>
<div class="displaymath" id="lin1">
  \begin{eqnarray} \label{lin1} \Psi _1(\textbf{w},\textbf{x},\phi ) & =&  D(\mathbf{w},\mathbf{x},\phi ;I)-\phi \big(a+b-\sum _{i=1}^nw_ix_i\big)\geq 0,\\ \Psi _2(\textbf{w},\textbf{x},\phi ) & =&  \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i)-D(\mathbf{w},\mathbf{x},\phi ;I) \geq 0,\\ \Psi _3(\textbf{w},\textbf{x},\phi ) & =&  \phi (a)+\phi (b)-\sum _{i=1}^nw_i\phi (x_i)-\phi \big(a+b-\sum _{i=1}^nw_ix_i\big)\geq 0. \end{eqnarray}
</div>
<p> Also, under the assumptions of Theorem <a href="#theniezref">7</a> using (<a href="#niezgodaineqref">6</a>) we define the functionals as follows: </p>
<div class="displaymath" id="a0000000067">
  \begin{eqnarray}  \Psi _4(\textbf{w},\textbf{X},\phi ) & =&  \tilde{D}(\mathbf{w},\mathbf{X},\phi ;I) -\phi \big(\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\big)\geq 0,\\ \Psi _5(\textbf{w},\textbf{X},\phi ) & =&  \sum _{j=1}^{m}\phi (a_j)-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\phi (x_{ij})- \tilde{D}(\mathbf{w},\mathbf{X},\phi ;I)\geq 0, \end{eqnarray}
</div>
<div class="equation" id="a0000000068">
<p>
  <div class="equation_content">
    \begin{equation}  \Psi _6(\textbf{w},\textbf{X},\phi ) = \sum _{j=1}^{m}\phi (a_j)-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\phi (x_{ij})- \phi \big(\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\big)\geq 0. \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p> Similarly, under the assumptions of Theorem <a href="#threffunt">20</a> using (<a href="#jensenmerreffunc">19</a>) we define the following functionals: </p>
<div class="displaymath" id="a0000000069">
  \begin{eqnarray}  \Psi _7(A,f,\phi ) & =&  \overline{D}(A,f,\phi ;E_1)-\phi (a+b-A(f))\geq 0,\\ \Psi _8(A,f,\phi ) & =&  \phi (a)+\phi (b)-A(\phi (f))- \overline{D}(A,f,\phi ;E_1)\geq 0,\\ \Psi _9(A,f,\phi ) & =&  \phi (a)+\phi (b)-A(\phi (f))-\phi (a+b-A(f))\geq 0. \end{eqnarray}
</div>
<p> Now we are in position to give mean value theorems for \(\Psi _j(.,.,\phi )\), \(j=1,2,\ldots ,9\). <div class="theorem_thmwrapper " id="meaneq1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">25</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\phi \in C^2([a,b])\), \(\textbf{x}=(x_1,...,x_n)\in [a,b]^n\) and \(\textbf{w}=(w_1,...,w_n)\) be \(n\)-tuple of positive real numbers such that \(\sum _{i=1}^nw_i=1\). Then there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000070">
  \begin{equation*} \label{meaneq1} \Psi _j(\textbf{w},\textbf{x},\phi )=\tfrac {\phi ”(c_j)}{2} \Psi _j(\textbf{w},\textbf{x},\phi _0), \, \, \, \hbox{where}\  \phi _0(x)=x^2; j=1,2,3. \end{equation*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000071">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Fix \(j=1,2,3 \).<br />Since the functions </p>
<div class="displaymath" id="a0000000072">
  \[ \phi _1=\tfrac {\Gamma }{2}x^2-\phi (x),\, \, \phi _2(x)=\phi (x)-\tfrac {\gamma }{2}x^2 \]
</div>
<p> are convex, where \(\Gamma =\max \limits _{x\in [a,b]}\phi ^{\prime \prime } (x)\) and \(\gamma =\min \limits _{x\in [a,b]}\phi ^{\prime \prime } (x)\), we have </p>
<div class="equation" id="profeq1">
<p>
  <div class="equation_content">
    \begin{equation} \label{profeq1} \Psi _j(\textbf{w},\textbf{x},\phi _1)\geq 0 \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<div class="equation" id="profeq2">
<p>
  <div class="equation_content">
    \begin{equation} \label{profeq2} \Psi _j(\textbf{w},\textbf{x},\phi _2)\geq 0. \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p> From (<a href="#profeq1">28</a>) and (<a href="#profeq2">29</a>) we get </p>
<div class="displaymath" id="a0000000073">
  \[ \tfrac {\gamma }{2}\Psi _j(\textbf{w},\textbf{x},\phi _0)\leq \Psi _j(\textbf{w},\textbf{x},\phi )\leq \tfrac {\Gamma }{2}\Psi _j(\textbf{w},\textbf{x},\phi _0). \]
</div>
<p> If \(\Psi _j(\textbf{w},\textbf{x},\phi _0)=0\) then there is nothing to prove. Suppose \(\Psi _j(\textbf{w},\textbf{x},\phi _0){\gt}0\). We have </p>
<div class="displaymath" id="a0000000074">
  \[ \gamma \leq \tfrac {2\Psi _j(\textbf{w},\textbf{x},\phi )}{\Psi _j(\textbf{w},\textbf{x},\phi _0)}\leq \Gamma . \]
</div>
<p> Hence, there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000075">
  \begin{equation*}  \Psi _j(\textbf{w},\textbf{x},\phi )=\tfrac {\phi ”(c_j)}{2} \Psi _j(\textbf{w},\textbf{x},\phi _0). \end{equation*}
</div>
<p> <div class="proof_wrapper" id="a0000000076">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="theorem_thmwrapper " id="meaneq2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">26</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\phi ,\psi \in C^2([a,b])\), \(\textbf{x}=(x_1,...,x_n)\in [a,b]^n\) and \(\textbf{w}=(w_1,...,w_n)\) be \(n\)-tuple of positive real numbers such that \(\sum _{i=1}^nw_i=1\). Then there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000077">
  \begin{equation*} \label{meaneq2} \tfrac { \Psi _j(\textbf{w},\textbf{x},\phi )}{ \Psi _j(\textbf{w},\textbf{x},\psi )}=\tfrac {\phi ”(c_j)}{\psi ”(c_j)},\, \,  j=1,2,3. \end{equation*}
</div>
<p> provided that the denominators are non-zero. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000078">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let us define </p>
<div class="displaymath" id="a0000000079">
  \[ g_j=a_j\phi -b_j\psi ,\, \, \, j=1,2,3, \]
</div>
<div class="displaymath" id="a0000000080">
  \[ \mbox{ where }\hspace{0cm} a_j=\Psi _j(\textbf{w},\textbf{x},\psi ),\, \, \; \; \; \; b_j=\Psi _j(\textbf{w},\textbf{x},\phi ). \]
</div>
<p> Obviously \(g_j\in C^2([a,b])\), by using Theorem <a href="#meaneq1">25</a> there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000081">
  \[ \left(\tfrac {a_j\phi ”(c_j)}{2}-\tfrac {b_j\psi ”(c_j)}{2}\right)\Psi _j(\textbf{w},\textbf{x},\phi _0)=0. \]
</div>
<p> Since \(\Psi _j(\textbf{w},\textbf{x},\phi _0)\neq 0\) (otherwise we have a contradiction with \(\Psi _j(\textbf{w},\textbf{x},\psi )\neq 0\) by Theorem <a href="#meaneq1">25</a>), we get </p>
<div class="displaymath" id="a0000000082">
  \begin{equation*}  \tfrac { \Psi _j(\textbf{w},\textbf{x},\phi )}{ \Psi _j(\textbf{w},\textbf{x},\psi )}=\tfrac {\phi ”(c_j)}{\psi ”(c_j)},\, \,  j=1,2,3. \end{equation*}
</div>
<p> <div class="proof_wrapper" id="a0000000083">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="theorem_thmwrapper " id="a0000000084">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">27</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\phi \in C^2([a,b])\), \(\textbf{a}=(a_1,...,a_m)\) with \(a_j\in [a,b]\), and \(\textbf{X}=(x_{ij})\) is a real \(n\times m\) matrix such that \(x_{ij}\in [a,b]\) for all \(i=1,\ldots ,n;\, \, j=1,\ldots ,m\) and \(\textbf{a}\) majorizes each row of \(\textbf{X}\). Then there exists \(c_k\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000085">
  \[ \Psi _k(\textbf{w},\textbf{X},\phi )=\tfrac {\phi ”(c_j)}{2} \Psi _k(\textbf{w},\textbf{X},\phi _0), \, \, \, \hbox{where}\  \phi _0(x)=x^2; k=4,5,6. \]
</div>

  </div>
</div> <div class="theorem_thmwrapper " id="a0000000086">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">28</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\phi ,\psi \in C^2([a,b])\). Suppose that \(\textbf{a}=(a_1,...,a_m)\) with \(a_j\in [a,b]\), and \(\textbf{X}=(x_{ij})\) is a real \(n\times m\) matrix such that \(x_{ij}\in [a,b]\) for all \(i=1,\ldots ,n;\, j=1,\ldots ,m\) and \(\textbf{a}\) majorizes each row of \(\textbf{X}\). Then there exists \(c_k\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000087">
  \[ \tfrac { \Psi _k(\textbf{w},\textbf{X},\phi )}{ \Psi _k(\textbf{w},\textbf{X},\psi )}=\tfrac {\phi ”(c_k)}{\psi ”(c_k)};\, \,  k=4,5,6, \]
</div>
<p> provided that the denominators are non-zero. </p>

  </div>
</div> <div class="theorem_thmwrapper " id="a0000000088">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">29</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose \(\phi \in C^2([a,b])\) and \(L\) satisfy properties \(L_1\), \(L_2\), on a nonempty set \(E\). Assume that \(A\) is an isotonic linear functional on \(L\) with \(A(\mathbf{1})\) = 1. Let \(f\in L\) be such that \(\phi (f)\in L\). Then there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000089">
  \begin{equation*}  \Psi _j(A,f,\phi )=\tfrac {\phi ”(c)}{2} \Psi _j(A,f,\phi _0), \, \, \, \hbox{where}\  \phi _0(x)=x^2; j=7,8,9. \end{equation*}
</div>

  </div>
</div> <div class="theorem_thmwrapper " id="a0000000090">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">30</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose \(\phi ,\psi \in C^2([a,b])\) and \(L\) satisfy properties \(L_1\), \(L_2\), on a nonempty set \(E\). Assume that \(A\) is an isotonic linear functional on \(L\) with \(A(\mathbf{1})\) = 1. Let \(f\in L\) be such that \(\phi (f),\psi (f)\in L\). Then there exists \(c_j\in [a,b]\) such that </p>
<div class="displaymath" id="a0000000091">
  \begin{equation*}  \tfrac { \Psi _j(A,f,\phi )}{ \Psi _j(A,f,\psi )}=\tfrac {\phi ”(c_j)}{\psi ”(c_j)},\, \,  j=7,8,9. \end{equation*}
</div>
<p> provided that the denominators are non-zero. </p>

  </div>
</div> </p>
<p><div class="rk_thmwrapper " id="genmeanrk">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">31</span>
  </div>
  <div class="rk_thmcontent">
  <p> If the inverse of \(\tfrac {\phi ”}{\psi ”}\) exists, then from the above mean value theorems we can give generalized means </p>
<div class="equation" id="genmean">
<p>
  <div class="equation_content">
    \begin{equation} \label{genmean} c_j=\left(\tfrac {\phi ”}{\psi ”}\right)^{-1}\left(\tfrac { \Psi _j(.,.,\phi )}{ \Psi _j(.,.,\psi )}\right),\, \, j=1,2,\ldots ,9.\hfil \qed \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000092">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">32</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#nexp" >10</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> A function \( \phi : J\rightarrow \mathbb {R} \) is <em>n-exponentially convex</em>  in the Jensen sense on the interval \(J\) if </p>
<div class="displaymath" id="a0000000093">
  \[  \sum _{k,l=1}^{n} \alpha _{k} \alpha _{l} \phi \left(\tfrac { x_{k} + x_{l}}{2} \right) \geq 0  \]
</div>
<p> holds for \( \alpha _{k} \in \mathbb {R} \) and \( x_{k} \in J \), \( k = 1,2,...,n \). </p>
<p>A function \( \phi : J\rightarrow \mathbb {R} \) is <em>n-exponentially convex</em>  if it is <em>n-exponentially convex</em>  in the Jensen sense and continuous on \(J\). </p>

  </div>
</div> </p>
<p><div class="rk_thmwrapper " id="a0000000094">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">33</span>
  </div>
  <div class="rk_thmcontent">
  <p>From the definition it is clear that \(1\)-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, \(n\)-exponentially convex functions in the Jensen sense are \(m\)-exponentially convex in the Jensen sense for every \(m \in \mathbb {N}, m \leq n\).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="proposition_thmwrapper " id="cr2">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">34</span>
  </div>
  <div class="proposition_thmcontent">
  <p> If \( \phi : J \rightarrow \mathbb {R} \) is an \(n\)-exponentially convex function, then the matrix \(\Big[\phi \left( \tfrac {x_{k}+x_{l}}{2} \right) \Big]_{k,l=1}^{m}\) is a positive semi-definite matrix for all \(m\in \mathbb {N},m\leq n\). Particularly, </p>
<div class="displaymath" id="a0000000095">
  \[  \det \left[\phi \left( \tfrac {x_{k}+x_{l}}{2} \right) \right]_{k,l=1}^{m} \geq 0  \]
</div>
<p> for all \( m \in \mathbb {N} \), \( m = 1,2,...,n \). </p>

  </div>
</div> <div class="definition_thmwrapper " id="a0000000096">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">35</span>
  </div>
  <div class="definition_thmcontent">
  <p>A function \( \phi : J \rightarrow \mathbb {R} \) is exponentially convex in the Jensen sense on \(I\) if it is \(n\)-exponentially convex in the Jensen sense for all \(n \in \mathbb {N}\). A function \( \phi : J \rightarrow \mathbb {R} \) is exponentially convex if it is exponentially convex in the Jensen sense and continuous. </p>

  </div>
</div> <div class="rk_thmwrapper " id="rk2exp">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">36</span>
  </div>
  <div class="rk_thmcontent">
  <p> It is easy to show that \( \phi : [a,b] \rightarrow \mathbb {R}^{+} \) is \(\log \)-convex in the Jensen sense if and only if </p>
<div class="displaymath" id="a0000000097">
  \[ \alpha ^2\phi (x)+2\alpha \beta \phi \left(\tfrac {x+y}{2}\right)+\beta ^2\phi (y)\geq 0 \]
</div>
<p> holds for every \(\alpha ,\beta \in \mbox{\black R}\) and \(x,y\in [a,b]\). It follows that a function is log-convex in the Jensen-sense if and only if it is \(2\)-exponentially convex in the Jensen sense. </p>
<p>Also, using basic convexity theory it follows that a function is \(\log \)-convex if and only if it is \(2\)-exponentially convex.<span class="qed">â–¡</span></p>

  </div>
</div> When dealing with functions with different degree of smoothness divided differences are found to be very useful. <div class="definition_thmwrapper " id="a0000000098">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">37</span>
  </div>
  <div class="definition_thmcontent">
  <p>The second order divided difference of a function \(\phi :[a,b]\linebreak \rightarrow \mbox{\black R}\) at mutually different points \(y_0,y_1,y_2\in [a,b]\) is defined recursively by </p>
<div class="displaymath" id="a0000000099">
  \[ [y_i;\phi ]=\phi (y_i),\, \, i=0,1,2 \]
</div>
<div class="displaymath" id="a0000000100">
  \[ [y_i,y_{i+1};\phi ]=\tfrac {\phi (y_{i+1})-\phi (y_i)}{y_{i+1}-y_i},\, \, i=0,1 \]
</div>
<div class="equation" id="divid">
<p>
  <div class="equation_content">
    \begin{equation} \label{divid} [y_0,y_{1},y_{2};\phi ]=\tfrac {[y_1,y_{2};\phi ]-[y_0,y_{1};\phi ]}{y_{2}-y_0}. \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>

  </div>
</div> <div class="rk_thmwrapper " id="2">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">38</span>
  </div>
  <div class="rk_thmcontent">
  <p>The value \([y_0,y_1,y_2;\phi ]\) is independent of the order of the points \(y_0,y_1\), and \(y_2\). By taking limits this definition may be extended to include the cases in which any two or all three points coincide as follows: \(\forall y_0,\, y_1,\, y_2 \in [a,b]\) </p>
<div class="displaymath" id="a0000000101">
  \begin{equation*} \label{divi} \lim \limits _{y_1 \rightarrow y_0 } [y_0,y_1,y_2;\phi ] =[y_0,y_0,y_2;\phi ]=\tfrac {f(y_2)-f(y_0)-{\phi ^{'}(y_0)}(y_2-y_0)}{{(y_2-y_0)}^2},\, \, \, \, y_2\neq y_0 \end{equation*}
</div>
<p> provided that \(\phi '\) exists, and furthermore, taking the limits \(y_i\rightarrow y_0, i=1,2\) in (<a href="#divid">31</a>), we get </p>
<div class="displaymath" id="a0000000102">
  \begin{equation*} \label{2} [y_0,y_0,y_0;\phi ]\, \,  =\,  \lim \limits _{y_i \rightarrow y_0 }[y_0,y_1,y_2;\phi ]=\tfrac {{\phi ^{''}(y_0)}}{2}\, \, \text{for}\, \, \,  i=1,2 \end{equation*}
</div>
<p> provided that \(\phi ^{''}\) exists on [a,b].<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>We use an idea from <span class="cite">
	[
	<a href="#Julinexp" >5</a>
	]
</span> to give an elegant method of producing an \(n\)-exponentially convex functions and exponentially convex functions applying the functionals \(\Psi _j(.,.,\phi )\), \(j=1,\ldots ,9\), on a given family with the same property. <div class="theorem_thmwrapper " id="nexpthem">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">39</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\Lambda =\{ \phi _t:t\in J\} \), where \(J\) is an interval in \(\mathbb {R}\), be a family of functions defined on an interval \([a,b]\), such that the function \(t\rightarrow [y_0,y_1,y_2;\phi _t]\) is \(n\)-exponentially convex in the Jensen sense on \(J\) for every three mutually different points \(y_0,y_1,y_2\in [a,b]\). Let \(\Psi _j(.,.,\phi _t)\) \((j=1,2,\ldots ,9)\) be linear functionals defined as in <span class="rm">(<a href="#lin1">19</a>)–(27)</span>. Then \(t\rightarrow \Psi _j(.,.,\phi _t)\) is an \(n\)-exponentially convex function in the Jensen sense on \(J\). If the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is continuous on \(J\), then it is \(n\)-exponentially convex on \(J\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000103">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Fix \(1\leq j \leq 9\).<br />Let us define the function </p>
<div class="displaymath" id="a0000000104">
  \[ \omega (y)=\sum _{k,l=1}^{n}b_{k}b_{l}\phi _{t_{kl}}(y), \]
</div>
<p> where \(t_{kl}=\tfrac {t_{k}+t_{l}}{2}\), \(t_{k}\in J,b_k\in \mathbb {R}\), \(k=1,2,...,n\).<br />Since the function \(t\rightarrow [y_0,y_1,y_2;\phi _t]\) is \(n\)-exponentially convex in the Jensen sense, we have </p>
<div class="displaymath" id="a0000000105">
  \[ [y_0,y_1,y_2;\omega ]=\sum _{k,l=1}^{n}b_{k}b_{l}[y_0,y_1,y_2;\phi _{t_{kl}}]\geq 0, \]
</div>
<p> which implies that \(\omega \) is a convex function on \([a,b]\) and therefore we have \(\Psi _j(.,.,\omega )\geq 0\); \(j=1,2,...,9\). Hence </p>
<div class="displaymath" id="a0000000106">
  \[ \sum _{k,l=1}^{n}b_{k}b_{l}\Psi _j(.,.,\phi _{t_{kl}})\geq 0. \]
</div>
<p> We conclude that the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is an \(n\)-exponentially convex function in the Jensen sense on \(J\). </p>
<p>If the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is continuous on \(J\), then it is \(n\)-exponentially convex on \(J\) by definition. <div class="proof_wrapper" id="a0000000107">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> As a consequence of the above theorem we can give the following corollary. <div class="cor_thmwrapper " id="corr6">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">40</span>
  </div>
  <div class="cor_thmcontent">
  <p> Let \(\Lambda =\{ \phi _t:t\in J\} \), where \(J\) is an interval in \(\mathbb {R}\), be a family of functions defined on an interval \([a,b]\), such that the function \(t\rightarrow [y_0,y_1,y_2;\phi _t]\) is exponentially convex in the Jensen sense on \(J\) for every three mutually different points \(y_0,y_1,y_2\in [a,b]\). Let \(\Psi _j(.,.,\phi _t)\) \((j=1,2,\ldots ,9)\) be linear functionals defined as in <span class="rm">(<a href="#lin1">19</a>)–(27)</span>. Then \(t\rightarrow \Psi _j(.,.,\phi _t)\) is an exponentially convex function in the Jensen sense on \(J\). If the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is continuous on \(J\), then it is exponentially convex on \(J\). </p>

  </div>
</div> </p>
<p><div class="cor_thmwrapper " id="a0000000108">
  <div class="cor_thmheading">
    <span class="cor_thmcaption">
    Corollary
    </span>
    <span class="cor_thmlabel">41</span>
  </div>
  <div class="cor_thmcontent">
  <p>Let \(\Lambda =\{ \phi _t:t\in J\} \), where \(J\) is an interval in \(\mathbb {R}\), be a family of functions defined on an interval \([a,b]\), such that the function \(t\rightarrow [y_0,y_1,y_2;\phi _t]\) is \(2\)-exponentially convex in the Jensen sense on \(J\) for every three mutually different points \(y_0,y_1,y_2\in [a,b]\). Let \(\Psi _j(.,.,\phi _t)\) \((j=1,2,\ldots ,9)\) be linear functionals defined as in <span class="rm">(<a href="#lin1">19</a>)–(27)</span>. Then the following statements hold: <em>(i)</em> If the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is continuous on \(J\), then it is \(2\)-exponentially convex on \(J\), and thus log convex on \(J\).<br /><em>(ii)</em> If the function \(t\rightarrow \Psi _j(.,.,\phi _t)\) is strictly positive and differentiable on \(J\), then for every \(s,t,u,v\in J\), such that \(s\leq u\) and \(t\leq v\), we have </p>
<div class="equation" id="meangenmono">
<p>
  <div class="equation_content">
    \begin{equation} \label{meangenmono} \mathfrak { B}_{s,t}(.,.,\Psi _j,\Lambda )\leq \mathfrak { B}_{u,v}(.,.,\Psi _j,\Lambda ) \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> where </p>
<div class="equation" id="meangen">
<p>
  <div class="equation_content">
    \begin{equation} \label{meangen} \mathfrak { B}^{j}_{s,t}(\Lambda )=\mathfrak { B}_{s,t}(.,.,\Psi _j,\Lambda )=\begin{cases}  \left(\tfrac {\Psi _j(.,.,\phi _s)}{\Psi _j(.,.,\phi _t)}\right)^{\tfrac {1}{s-t}}, & s\neq t,\\ \exp \left(\tfrac {\tfrac {d}{ds}\Psi _j(.,.,\phi _s)}{\Psi _j(.,.,\phi _s)}\right), & s=t, \end{cases} \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<p> for \(\phi _s,\phi _t\in \Lambda \). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000109">
  <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>
<ul class="itemize">
  <li><p>See Remark <a href="#rk2exp">36</a> and Theorem <a href="#nexpthem">39</a>. </p>
</li>
  <li><p>From the definition of convex function \(\phi \), we have the following inequality <span class="cite">
	[
	<a href="#redbook" >11</a>
	, 
	p.2
	]
</span> </p>
<div class="equation" id="a5">
<p>
  <div class="equation_content">
    \begin{equation} \label{a5} \tfrac {\phi \left(s\right)\, -\, \phi \left(t\right)}{s\, -\, t}\, \leq \,  \tfrac {\phi \left(u\right)\, -\, \phi \left(v\right)}{u\, -\, v}, \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> \( \forall \,  s, t,u,v \in J\) such that \(s\leq u,\, t\leq v,\, s\neq t,\, u\neq v\). <br />Since by (i), \( \Psi _j(.,.,\phi _s) \) is \(\log \)-convex, so set \(\phi (x)=\ln \Psi _j(.,.,\phi _x)\) in (<a href="#a5">36</a>) we have </p>
<div class="equation" id="eq7">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq7} \tfrac {\ln \Psi _j(.,.,\phi _s)\, -\ln \Psi _j(.,.,\phi _t)}{s-t} \, \leq \,  \tfrac {\ln \Psi _j(.,.,\phi _u)-\ln \Psi _j(.,.,\phi _v)}{u-v} \end{equation}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p> for \(s\leq u,\, t\leq v,\, s\neq t,\, u\neq v\), which equivalent to (<a href="#meangenmono">32</a>). The cases for \(s= t,\, u= v\) follow from (<a href="#a5">36</a>) by taking limit. </p>
</li>
</ul>
<p> <div class="proof_wrapper" id="a0000000110">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="rk_thmwrapper " id="a0000000111">
  <div class="rk_thmheading">
    <span class="rk_thmcaption">
    Remark
    </span>
    <span class="rk_thmlabel">42</span>
  </div>
  <div class="rk_thmcontent">
  <p>In <span class="cite">
	[
	<a href="#matlobpec" >1</a>
	]
</span> authors gave related results for the Jensen Mercer inequality.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000112">6 Examples</h1>
<p> In this section we will vary on choice of family of functions in order to give some examples of exponentially convex functions and to construct some means in the same way as given in <span class="cite">
	[
	<a href="#Julinexp" >5</a>
	]
</span> and <span class="cite">
	[
	<a href="#nexp" >10</a>
	]
</span>. For simplicity we assume that \(J(\textbf{a},X,\textbf{w})=\sum _{j=1}^{m}a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\). Let \(\phi _t\) be any function, \(t\in J\) where \(J\) is an interval in \(\mbox{\black R}\), we apply the conditions: </p>
<div class="displaymath" id="a0000000113">
  \[ \lim _{t\rightarrow t_0}A(\phi _t)=A(\lim _{t\rightarrow t_0}\phi _t), \]
</div>
<div class="displaymath" id="a0000000114">
  \[ \lim _{t\rightarrow t_0}\tfrac {A(\phi _{t+\Delta t})-A(\phi _t)}{\Delta t}=A\left(\lim _{t\rightarrow t_0}\tfrac {\phi _{t+\Delta t}-\phi _t}{\Delta t}\right). \]
</div>
<p> <div class="ex_thmwrapper " id="l2.101">
  <div class="ex_thmheading">
    <span class="ex_thmcaption">
    Example
    </span>
    <span class="ex_thmlabel">43</span>
  </div>
  <div class="ex_thmcontent">
  <p> Let </p>
<div class="displaymath" id="a0000000115">
  \begin{equation*}  \Lambda _1= \{ \psi _t:\mathbb {R}\rightarrow [0,\infty ):\,  t\in \mathbb {R}\}  \end{equation*}
</div>
<p> be the family of functions defined by </p>
<div class="displaymath" id="a0000000116">
  \begin{equation*} \label{l2.101} \psi _{t}(x)\, =\, \left\{  \begin{array}{ll} \tfrac {1}{t^{2}}\,  {\rm e}^{tx}, &  \hbox{t\, $\neq $ 0}, \\ \tfrac {1}{2}\, x^{2}, &  \hbox{t\, = 0.} \end{array} \right. \end{equation*}
</div>
<p> Since, \(\psi _{t}(x)\) is a convex function on \(\mathbb {R}\) and \(\psi ''_{t}(x)\) is exponentially convex function <span class="cite">
	[
	<a href="#Julinexp" >5</a>
	]
</span>, using analogous arguing as in the proof of Theorems <a href="#nexpthem">39</a> we have that \(t\mapsto [y_0,y_1,y_2;\psi _t]\) is exponentially convex (and so exponentially convex in the Jensen sense). Using Corollary <a href="#corr6">40</a> we conclude that \(t\mapsto \Psi _j(.,.,\psi _t);\, \, j=1,...,9\) are exponentially convex in the Jensen sense. It is easy to see that these mappings are continuous, so they are exponentially convex.<br />Assume that \(t\mapsto \Psi _j(.,.,\psi _t){\gt}0\) \((j=1,2,...,9)\). By using this family of convex functions in (<a href="#genmean">30</a>) for \(j=1,2,\ldots ,9\), we obtain the following means: </p>
<div class="displaymath" id="a0000000117">
  \begin{eqnarray*}  \Gamma ^{j}_{s,t}\, \,  =\,  \left\{  \begin{array}{ll} \tfrac {1}{s-t}\ln \left(\tfrac { \Psi _j(.,., \psi _s)}{\Psi _j(.,., \psi _t)}\right),\, \, \, & s\neq t, \\ \tfrac { \Psi _j(.,.,id. \psi _s)}{\Psi _j(.,., \psi _s)}-\tfrac {2}{s} , \, \, \,  &  s=t\neq 0, \\ \tfrac { \Psi _j(.,.,id. \psi _0)}{3\Psi _j(.,., \psi _0)},\, \, \, & s=t=0. \end{array} \right. \end{eqnarray*}
</div>
<p> In particular for \(j=6\) we have </p>
<div class="displaymath" id="a0000000118">
  \begin{eqnarray*}  \Gamma ^{6}_{s,t}& =& \tfrac {1}{s-t}\ln \left(\tfrac {t^2\big(\sum _{j=1}^{m}e^{sa_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{sx_{ij}}- e^{sJ(\textbf{a},X,\textbf{w})}\big) } {s^2\big(\sum _{j=1}^{m}e^{ta_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{tx_{ij}}- e^{tJ(\textbf{a},X,\textbf{w})}\big) }\right),\, s\neq t;\;  s,t\neq 0,\\ \Gamma ^{6}_{s,s}& =& \tfrac {\sum _{j=1}^{m}a_je^{sa_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}e^{sx_{ij}} -J(\textbf{a},X,\textbf{w})e^{sJ(\textbf{a},X,\textbf{w})} } {\sum _{j=1}^{m}e^{sa_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{sx_{ij}}- e^{sJ(\textbf{a},X,\textbf{w})} }-\tfrac {2}{s},\, s\neq 0,\\ \Gamma ^{6}_{s,0}& =& \tfrac {1}{s}\ln \left(\tfrac {2\big(\sum _{j=1}^{m}e^{sa_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{sx_{ij}}- e^{sJ(\textbf{a},X,\textbf{w})}\big) } {s^2\big(\sum _{j=1}^{m}a_j^{2}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}^{2}- J^{2}(\textbf{a},X,\textbf{w})\big) }\right),\,  s\neq 0,\\ \Gamma ^{6}_{0,0}& =& \tfrac {\sum _{j=1}^{m}a_j^{3}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{3}_{ij} -J^{3}(\textbf{a},X,\textbf{w}) } {3(\sum _{j=1}^{m}a_j^{2}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{2}_{ij} -J^{2}(\textbf{a},X,\textbf{w}))} .\\ \end{eqnarray*}
</div>
<p> Since \(\Gamma ^{j}_{s,t}= \ln \mathfrak {B}^{j}_{s,t}(\Lambda _1)\) \((j=1,2,...,9)\), so by (<a href="#meangenmono">32</a>) these means are monotonic.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="ex_thmwrapper " id="ex1">
  <div class="ex_thmheading">
    <span class="ex_thmcaption">
    Example
    </span>
    <span class="ex_thmlabel">44</span>
  </div>
  <div class="ex_thmcontent">
  <p> Let </p>
<div class="displaymath" id="a0000000119">
  \begin{equation*}  \Lambda _2=\{ \varphi _t:(0,\infty )\rightarrow \mathbb {R}:t\in \mathbb {R}\}  \end{equation*}
</div>
<p> be the family of functions defined by<br /></p>
<div class="displaymath" id="a0000000120">
  \begin{equation*}  \varphi _t(x)\, =\, \left\{  \begin{array}{rcl} \tfrac {x^{t}}{t(t-1)}, &  \hbox{t $\neq $0,1}, \\ -\ln x, &  \hbox{t=0}, \\ x\ln x, &  \hbox{t=1}. \end{array} \right. \end{equation*}
</div>
<p> Since \(\varphi _t(x)\) is a convex function for \(x\in \mathbb {R}^{+}\) and \(t\rightarrow \varphi ''_{t}(x)\) is exponentially convex, so by the same arguments given in previous example we conclude that \(\Psi _j(.,.,\varphi _t);\, \, j=1,...,9\) are exponentially convex. We assume that \([a,b]\subset \mathbb {R}^+\) and \(\Psi _j(.,.,\varphi _t){\gt}0 (j=1,...,9)\). By using this family of convex functions in (<a href="#genmean">30</a>) for \(j=1,2,\ldots ,9\) we have the following means: </p>
<div class="displaymath" id="a0000000121">
  \begin{eqnarray*}  \tilde{\Gamma }^{j}_{s,t} =\,  \left\{  \begin{array}{ll} \left(\tfrac { \Psi _j(.,., \varphi _s)}{\Psi _j(.,., \varphi _t)}\right)^{\tfrac {1}{s-t}},& s\neq t, \\ \exp \Big(\tfrac {1-2s}{s(s-1)}-\tfrac { \Psi _j(.,.,\varphi _0 \varphi _s)}{\Psi _j(.,., \varphi _s)}\Big),&  s = t\neq 0,1, \\ \exp \Big(1-\tfrac { \Psi _j(.,.,{\varphi _0}^2)}{2\Psi _j(.,., \varphi _0)}\Big),& s=t=0, \\ \exp \Big(-1-\tfrac { \Psi _j(.,.,\varphi _0 \varphi _1)}{2\Psi _j(.,., \varphi _1)}\Big),& s=t=1. \end{array} \right. \end{eqnarray*}
</div>
<p> In particular for \(j=6\) we have </p>
<div class="displaymath" id="a0000000122">
  \begin{align*} & \tilde{\Gamma }^{6}_{s,t}\! =\! \left(\tfrac {t(t-1)}{s(s-1)}.\tfrac {\sum _{j=1}^{m}a_j^{s}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{s}_{ij}- J^{s}(\textbf{a},X,\textbf{w}) } {\sum _{j=1}^{m}a_j^{t}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{t}_{ij}- J^{t}(\textbf{a},X,\textbf{w}) }\right)^\frac {1}{s-t},s\neq t;\;  s,t\neq 0,1,\\ & \tilde{\Gamma }^{6}_{s,s}=\exp \left(\tfrac {1-2s}{s(s-1)} \right.\\ & \quad \  \quad \left.-\tfrac {\sum _{j=1}^{m}\ln a_j a_j^{s}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\ln x_{ij}x^{s}_{ij}- \ln J(\textbf{a},X,\textbf{w}) J^{s}(\textbf{a},X,\textbf{w}) } {\sum _{j=1}^{m}a_j^{s}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{s}_{ij}- J^{s}(\textbf{a},X,\textbf{w}) } \right),s\neq 0,1,\\ & \tilde{\Gamma }^{6}_{s,0}=\left(\tfrac {\sum _{j=1}^{m}a_j^{s}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{s}_{ij}- J^{s}(\textbf{a},X,\textbf{w}) } {s(1-s)\sum _{j=1}^{m}\ln a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\ln x_{ij}-\ln J(\textbf{a},X,\textbf{w}) }\right)^\frac {1}{s}, s\neq 0,\\ & \tilde{\Gamma }^{6}_{s,1}=\left(\tfrac {\sum _{j=1}^{m}a_j^{s}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{s}_{ij}- J^{s}(\textbf{a},X,\textbf{w}) } {s(s-1)\sum _{j=1}^{m}a_j\ln a_j-\sum _{j=1}^{m-1}\sum _{i=1}^nw_i x_{ij}\ln x_{ij}-J(\textbf{a},X,\textbf{w})\ln J(\textbf{a},X,\textbf{w}) }\right)^\frac {1}{s-1} s\neq 0,\\ & \tilde{\Gamma }^{6}_{0,0}=\exp \left(1-\tfrac {\sum _{j=1}^{m}\ln ^{2} a_j -\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\ln ^{2} x_{ij}- \ln ^{2}J(\textbf{a},X,\textbf{w}) } {2(\sum _{j=1}^{m}\ln a_j -\sum _{j=1}^{m-1}\sum _{i=1}^nw_i\ln x_{ij}- \ln J(\textbf{a},X,\textbf{w})) }\right),\\ & \tilde{\Gamma }^{6}_{1,1}=\exp \left(-1-\tfrac {\sum _{j=1}^{m}a_j\ln ^{2} a_j -\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\ln ^{2}x_{ij}- J(\textbf{a},X,\textbf{w})\ln ^{2}J(\textbf{a},X,\textbf{w}) } {2(\sum _{j=1}^{m}a_j\ln a_j -\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}\ln x_{ij}- J(\textbf{a},X,\textbf{w})\ln J(\textbf{a},X,\textbf{w})) }\right). \end{align*}
</div>
<p> Since \(\tilde{\Gamma }^{j}_{s,t}= \mathfrak {B}^{j}_{s,t}(\Lambda _2)\) \((j=1,2,...,9)\), so by (<a href="#meangenmono">32</a>) these means are monotonic.<span class="qed">â–¡</span></p>

  </div>
</div> <div class="ex_thmwrapper " id="a0000000123">
  <div class="ex_thmheading">
    <span class="ex_thmcaption">
    Example
    </span>
    <span class="ex_thmlabel">45</span>
  </div>
  <div class="ex_thmcontent">
  <p>Let </p>
<div class="displaymath" id="a0000000124">
  \begin{equation*}  \Lambda _3= \{ \theta _t:(0,\infty )\rightarrow (0,\infty ):\,  t\in (0,\infty )\}  \end{equation*}
</div>
<p> <br />be the family of functions defined by </p>
<div class="displaymath" id="a0000000125">
  \begin{eqnarray*}  \theta _t(x)=\tfrac {{\rm e}^{-x\sqrt{t}}}{t}. \end{eqnarray*}
</div>
<p> Since \(t\rightarrow \tfrac {d^2}{dx^2}\theta _{t}(x)= {\rm e}^{-x\sqrt{t}}\) is exponentially convex, being the Laplace transform of a non-negative function <span class="cite">
	[
	<a href="#Julinexp" >5</a>
	]
</span>, so by same argument given in Example <a href="#l2.101">43</a> we conclude that \(\Psi _j(.,.,\theta _t);\, \; j=1,...,9\) are exponentially convex. We assume that \([a,b]\, \subset \, \mathbb {R}^+\) and \(\Psi _j(.,.,\theta _t)\, {\gt}\, 0 (j=1,...,9)\). For this family of convex functions \(\mathfrak { B}^{j}_{s,t}(\Lambda _3)\) \((j=1,2,\ldots ,9)\) from (<a href="#meangen">33</a>) become </p>
<div class="displaymath" id="a0000000126">
  \begin{eqnarray*}  \mathfrak { B}^{j}_{s,t}(\Lambda _3)\, \,  =\,  \left\{  \begin{array}{ll} \left(\tfrac { \Psi _j(.,., \theta _s)}{\Psi _j(.,., \theta _t)}\right)^{\tfrac {1}{s-t}},& s\neq t, \\ \exp \Big(-\tfrac {\Psi _j(.,.,id. \theta _s)}{2\sqrt{s}(\Psi _j(.,., \theta _s))}-\tfrac {1}{s}\Big),& s=t. \end{array} \right. \end{eqnarray*}
</div>
<p> In particular for \(j=6\) we have </p>
<div class="displaymath" id="a0000000127">
  \begin{align*} & \mathfrak { B}^{6}_{s,t}(\Lambda _3)=\left(\tfrac {t}{s}.\tfrac {\sum _{j=1}^{m}e^{-a_j\sqrt{s}}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{-x_{ij}\sqrt{s}}- e^{-J(\textbf{a},X,\textbf{w})\sqrt{s}} } {\sum _{j=1}^{m}e^{-a_j\sqrt{t}}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{-x_{ij}\sqrt{t}}- e^{-J(\textbf{a},X,\textbf{w})\sqrt{t}} }\right)^\frac {1}{s-t},\, s\neq t,\\ & \mathfrak { B}^{6}_{s,s}(\Lambda _3)=\\ & =\exp \left(-\tfrac {1}{2\sqrt{s}}\tfrac {\sum _{j=1}^{m}a_je^{-a_j\sqrt{s}}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}e^{-x_{ij}\sqrt{s}}- J(\textbf{a},X,\textbf{w})e^{-J(\textbf{a},X,\textbf{w})\sqrt{s}} } {\sum _{j=1}^{m}e^{-a_j\sqrt{s}}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ie^{-x_{ij}\sqrt{s}}- e^{-J(\textbf{a},X,\textbf{w})\sqrt{s}} }-\tfrac {1}{s}\right). \end{align*}
</div>
<p> Monotonicity of \(\mathfrak { B}^{j}_{s,t}(\Lambda _3)\) follows from (<a href="#meangenmono">32</a>). By (<a href="#genmean">30</a>) </p>
<div class="displaymath" id="a0000000128">
  \[ \bar{\Gamma }^{j}_{s,t}= -(\surd s+\surd t)\ln \mathfrak {B}^{j}_{s,t}(\Lambda _3) \; \; \; (j=1,2,...,9) \]
</div>
<p> defines a class of means.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="ex_thmwrapper " id="a0000000129">
  <div class="ex_thmheading">
    <span class="ex_thmcaption">
    Example
    </span>
    <span class="ex_thmlabel">46</span>
  </div>
  <div class="ex_thmcontent">
  <p>Let </p>
<div class="displaymath" id="a0000000130">
  \begin{equation*}  \Lambda _4=\{ \phi _t:(0,\infty )\rightarrow (0,\infty ): t\in (0,\infty )\}  \end{equation*}
</div>
<p> be the family of functions defined by </p>
<div class="displaymath" id="a0000000131">
  \begin{eqnarray*}  \phi _t(x)= \left\{ \begin{array}{ll} \tfrac {t^{-x}}{(\ln t)^2},& t\neq 1, \\ \tfrac {x^{2}}{2},& t=1. \end{array} \right. \end{eqnarray*}
</div>
<p> Since \({\tfrac {d^2}{dx^2}\phi _{t}(x)}= t^{-x}={\rm e}^{-xlnt}{\gt}0\), for \(x{\gt}0\), so by same argument given in Example <a href="#l2.101">43</a> we conclude that \(t\rightarrow \Psi _j(.,.,\phi _t);\, \,  j=1,...,9\) are exponentially convex. We assume that \([a,b]\subset \mathbb {R}^+\) and \(\Psi _j(.,.,\phi _t){\gt}0 (j=1,...,9)\). For this family of convex functions \(\mathfrak { B}^{j}_{s,t}( \Lambda _4)\) \((j=1,2,\ldots ,9)\) from (<a href="#meangen">33</a>) become </p>
<div class="displaymath" id="a0000000132">
  \begin{eqnarray*}  \mathfrak { B}^{j}_{s,t}( \Lambda _4) = \left\{  \begin{array}{ll} \left(\tfrac { \Psi _j(.,., \phi _s)}{\Psi _j(.,., \phi _t)}\right)^{\tfrac {1}{s-t}},& s\neq t, \\ \exp \Big(-\tfrac {\Psi _j(.,.,id. \phi _s)}{s\Psi _j(.,., \phi _s)}-\tfrac {2}{s\ln s}\Big),& s=t\neq 1, \\ \exp \Big(\tfrac {1}{3}\tfrac { \Psi _j(.,.,id. \phi _1)}{\Psi _j(.,., \phi _1)}\Big),& s=t=1, \end{array} \right. \end{eqnarray*}
</div>
<p> In particular for \(j=6\) we have </p>
<div class="displaymath" id="a0000000133">
  \begin{align*}  \mathfrak { B}^{6}_{s,t}(\Lambda _4)& =\left(\tfrac {(\ln t)^2}{(\ln s)^2}.\tfrac {\sum _{j=1}^{m}s^{-a_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_is^{-x_{ij}}- s^{-J(\textbf{a},X,\textbf{w})} } {\sum _{j=1}^{m}t^{-a_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_it^{-x_{ij}}- t^{-J(\textbf{a},X,\textbf{w})} }\right)^\frac {1}{s-t},s\neq t,\\ \mathfrak { B}^{6}_{s,s}( \Lambda _4)& = \exp \left(-\tfrac {1}{s}\tfrac {\sum _{j=1}^{m}a_js^{-a_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}s^{-x_{ij}}- J(\textbf{a},X,\textbf{w})s^{-J(\textbf{a},X,\textbf{w})} } {\sum _{j=1}^{m}s^{-a_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_is^{-x_{ij}}- s^{-J(\textbf{a},X,\textbf{w})} }\right.\\ & \quad \quad \quad \left.-\tfrac {2}{s\ln s}\right),s\neq 1,\\ \mathfrak { B}^{6}_{s,1}( \Lambda _4)& =\left(\tfrac {2(\sum _{j=1}^{m}s^{-a_j}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_is^{-x_{ij}}- s^{-J(\textbf{a},X,\textbf{w})} )} {(\ln s)^{2}[\sum _{j=1}^{m}a_j^{2}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}^{2}- J^{2}(\textbf{a},X,\textbf{w}) ]}\right)^{\tfrac {1}{s-1}},\\ \mathfrak { B}^{6}_{1,1}( \Lambda _4)& =\tfrac {\sum _{j=1}^{m}a_j^{3}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix_{ij}^{3}- J^{3}(\textbf{a},X,\textbf{w}) } {3(\sum _{j=1}^{m}a_j^{2}-\sum _{j=1}^{m-1}\sum _{i=1}^nw_ix^{2}_{ij}- J^{2}(\textbf{a},X,\textbf{w})) }. \end{align*}
</div>
<p> Monotonicity of \(\mathfrak { B}^{j}_{s,t}( \Lambda _4)\) follows from (<a href="#meangenmono">32</a>). By (<a href="#genmean">30</a>) </p>
<div class="displaymath" id="a0000000134">
  \[ \hat{\Gamma }^{j}_{s,t}= -L(s,t)\ln \mathfrak {B}^{j}_{s,t}(\Lambda _4) \; \; \; (j=1,2,...,9) \]
</div>
<p> defines a class of means, where \(L(s,t)\) is Logarithmic mean defined as: </p>
<div class="displaymath" id="a0000000135">
  \begin{equation*}  L(s,t)\, =\, \left\{  \begin{array}{rcl} \tfrac {s-t}{\ln s-\ln t }, &  \hbox{s $\neq $ t}, \\ s, &  \hbox{s=t}. \end{array} \right. \end{equation*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="matlobpec">1</a></dt>
  <dd><p><i class="sc">M. Anwar</i> and <i class="sc">J. Pečarić</i>, <i class="it">Cauchy means of Mercer’s type</i>, Utilitas Mathematica, <b class="bf">84</b>, pp.&#160;201–208, 2011. </p>
</dd>
  <dt><a name="Mitrinovic">2</a></dt>
  <dd><p><i class="sc">P.S. Bullen, D.S. Mitrinović</i> and <i class="sc">P.M. Vasić</i>, <i class="it">Means and Their Inequalities</i>, Reidel, Dordrecht, 1988. </p>
</dd>
  <dt><a name="Cheung">3</a></dt>
  <dd><p><i class="sc">W.S. Cheung, A. Matković</i> and <i class="sc">J. Pečarić</i>, <i class="it">A variant of Jessen’s inequality and generalized means</i>, JIPAM, <b class="bf">7</b>(1), Article 10, 2006. </p>
</dd>
  <dt><a name="dragnerefjen">4</a></dt>
  <dd><p><i class="sc">S.S. Dragomir</i>, <i class="it">A new refinement of Jensen’s inequality in linear spaces with applications</i>, Mathematical and Computer Modelling, <b class="bf">52</b>, pp.&#160;1497-1505, 2010. </p>
</dd>
  <dt><a name="Julinexp">5</a></dt>
  <dd><p><i class="sc">J. Jakšetić</i> and <i class="sc">J. Pečarić</i>, <i class="it">Exponential Convexity Method</i>, J. Convex Anal., to appear. </p>
</dd>
  <dt><a name="marshal">6</a></dt>
  <dd><p><i class="sc">A.W. Marshall, I. Olkin</i> and <i class="sc">B.C. Arnold</i>, <i class="it">Inequalities: Theory of majorization and its applications (Second Edition)</i>, Springer Series in Statistics, New York 2011. </p>
</dd>
  <dt><a name="pecaricref">7</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no1-art10"> <i class="sc">A. Matković</i> and <i class="sc">J. Pečarić</i>, <i class="it">Refinements of the Jensen-Mercer inequality for index set functions with applications</i>, Revue d’analyse numérique et de théorie de l’approximation, <b class="bf">35</b>, no. 1, pp.&#160;71-82, 2006. <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="mercer">8</a></dt>
  <dd><p><i class="sc">A.McD. Mercer</i>, <i class="it">A variant of Jensen’s inequality</i>, J. Ineq. Pure and Appl. Math., <b class="bf">4</b>(4), 2003, Article 73. </p>
</dd>
  <dt><a name="niezgodamercer">9</a></dt>
  <dd><p><i class="sc">M. Niezgoda</i>, <i class="it">A generalization of Mercer’s result on convex functions</i>, Nonlinear Anal., <b class="bf">71</b>, pp.&#160; 2771–2779, 2009. </p>
</dd>
  <dt><a name="nexp">10</a></dt>
  <dd><p><i class="sc">J. Pečarić</i> and <i class="sc">J. Perić</i>, <i class="it">Improvement of the Giaccardi and the Petrović inequality and related Stolarsky type means</i>, An. Univ. Craiova Ser. Mat. Inform., to appear. </p>
</dd>
  <dt><a name="redbook">11</a></dt>
  <dd><p><i class="sc">J. Pečarić, F. Proschan</i> and <i class="sc">Y.L. Tong</i>, <i class="it">Convex functions, Partial Orderings and Statistical Applications</i>, Academic Press, New York, 1992. </p>
</dd>
</dl>


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