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<title>\(h\)-Strongly \(E\)-Convex Functions: \(h\)-Strongly \(E\)-Convex Functions</title>
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<h1>\(h\)-Strongly \(E\)-Convex Functions</h1>
<p class="authors">
<span class="author">Daniela Marian\(^{\ast }\)</span>
</p>
<p class="date">January 12, 2011.</p>
</div>
<p>\(^{\ast }\)Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Constantin Daicoviciu, no. 15, 400020 Cluj-Napoca, Romania, e-mail: <span class="ttfamily">daniela.marian@math.utcluj.ro</span>. </p>

<div class="abstract"><p> Starting from strongly \(E\)-convex functions introduced by E. A. <br />Youness, and T. Emam, from \(h\)-convex functions introduced by S. Varošanec and from the more general concept of \(h\)-convex functions introduced by A. Házy we define and study \(h\)-strongly \(E\)-convex functions. We study some properties of them. </p>
<p><b class="bf">MSC.</b> 26B25. </p>
<p><b class="bf">Keywords.</b> Strongly \(E\)-convex sets, strongly \(E\)-convex functions, \(h\)-convex functions, \(h\)-strongly \(E\)-convex functions. </p>
</div>
<h1 id="a0000000002">1 Preliminary notions and results</h1>
<p>The concepts of \(E\)-convex sets and \(E\)-convex functions were introduced by Youness in <span class="cite">
	[
	<a href="#You99" >8</a>
	]
</span>. Subsequently, Chen introduced a new concept of semi-\(E\)-convex functions in <span class="cite">
	[
	<a href="#Che02" >2</a>
	]
</span>. Based upon these approaches, in <span class="cite">
	[
	<a href="#You05a" >9</a>
	]
</span> Youness and Emam introduced the concepts of strongly \(E\)-convex sets and strongly \(E\)-convex functions. We firstly recall the definitions of convex sets, convex functions, \(E\)-convex sets and \(E\)-convex functions then of strongly \(E\)-convex sets and strongly \(E\)-convex functions and finally the definitions of \(h\)-convex functions, in the sense of Varošanec <span class="cite">
	[
	<a href="#Var07" >7</a>
	]
</span> and Házy <span class="cite">
	[
	<a href="#Haz11" >4</a>
	]
</span>. </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000003">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p>A set \(A\subset \mathbb {R}^{n}\, \) is called convex if \(\lambda x+\left( 1-\lambda \right) y\in A\), for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000004">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">2</span>
  </div>
  <div class="definition_thmcontent">
  <p>A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called convex on a convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied: </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq \lambda f(x)+\left( 1-\lambda \right) f(y). \end{equation*}
</div>

  </div>
</div> </p>
<p>We consider a function \(E:\mathbb {R}^{n}\to \mathbb {R}^{n}\). </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000006">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">3</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#You99" >8</a>
	]
</span>A set \(A\subset \mathbb {R}^{n}\, \) is called \(E\)-convex if \(\lambda E(x)+\left( 1-\lambda \right) E(y)\in A\), for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] .\) </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000007">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">4</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#You99" >8</a>
	]
</span>A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called \(E\)-convex on an \(E\)-convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied: </p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  f\left( \lambda E(x)+\left( 1-\lambda \right) E(y)\right) \leq \lambda f(E(x))+\left( 1-\lambda \right) f(E(y)). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000009">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">5</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#You05a" >9</a>
	]
</span>A set \(A\subset \mathbb {R}^{n}\, \) is called strongly \(E\)-convex if </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) \in A, \end{equation*}
</div>
<p> for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000011">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">6</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#You05a" >9</a>
	]
</span>A function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is called strongly \(E\)-convex on a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n}\) if for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), the following inequality is satisfied: </p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) ) \leq \lambda f(E(x))+\left( 1-\lambda \right) f(E(y)). \end{equation*}
</div>

  </div>
</div> </p>
<p>In the following lines we recall the definition of \(h\)-convex functions introduced in <span class="cite">
	[
	<a href="#Var07" >7</a>
	]
</span> by S. Varošanec. </p>
<p>We consider I and J intervals in \(\mathbb {R}\), \(\left( 0,1\right) \subseteq J\) and the real non-negative functions \(h:J\to \mathbb {R},\) \(f:I\to \mathbb {R},h\neq 0.\) </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000013">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">7</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#Var07" >7</a>
	]
</span>The function \(f:I\to \mathbb {R}\  \)is called \(h\)-convex on I or is said to belong to the class \(SX\left( h,I\right) \) if for every pair of points \(x,y\in I\) and every \(\lambda \in \left( 0,1\right) \), the following inequality is satisfied: </p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq h(\lambda )f(x)+h\left( 1-\lambda \right) f(y). \end{equation*}
</div>

  </div>
</div> </p>
<p>In <span class="cite">
	[
	<a href="#BomVar09" >1</a>
	]
</span> Bombardelli and Varošanec omitted the assumption that f and h are non-negative. We recall now the definitions of \(h\)-convex functions introduced in <span class="cite">
	[
	<a href="#Haz11" >4</a>
	]
</span> by A. Házy. </p>
<p>Let X be a real (complex) linear space and \(A \subset X\) nonempty, convex, open. Let \(h:[0,1]\to \mathbb {R},\) \(f:A\to \mathbb {R}\). </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000015">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">8</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="cite">
	[
	<a href="#Haz11" >4</a>
	]
</span>The function \(f:A\to \mathbb {R}\  \)is called \(h\)-convex on A if for every pair of points \(x,y\in A\) and every \(\lambda \in \left[ 0,1\right] \), the following inequality is satisfied: </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  f\left( \lambda x+\left( 1-\lambda \right) y\right) \leq h(\lambda )f(x)+h\left( 1-\lambda \right) f(y). \end{equation*}
</div>

  </div>
</div> </p>
<h1 id="a0000000017">2 Properties of \(\lowercase {h}\)-Strongly \(E\)-convex Functions</h1>
<p>Starting from strongly \(E\)-convex functions and from \(h\)-convex functions in the sense of Házy we define and study \(h\)-strongly \(E\)-convex functions. </p>
<p>In the following lines we consider a map \(E:\mathbb {R}^{n}\to \mathbb {R}^{n}\) and a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n}.\) We also consider the functions \(h:[0,1]\to \mathbb {R},\) \(f:A\to \mathbb {R}\). </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000018">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">9</span>
  </div>
  <div class="definition_thmcontent">
  <p>A function \(f:A\to \mathbb {R}\) is called \(h\)-strongly \(E\)-convex on \(A\) if for every pair of points \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), the following inequality is satisfied:</p>
<div class="equation" id="hstrEcv">
<p>
  <div class="equation_content">
    \begin{equation}  f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\leq h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right). \label{hstrEcv} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="a0000000019">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f:A\to \mathbb {R}\) is \(h\)-strongly \(E\)-convex on \(A\) and \(h(0)=0\) then </p>
<div class="equation" id="hstrEcv1">
<p>
  <div class="equation_content">
    \begin{equation}  f\left( \alpha x+E\left( x\right) \right) \leq h\left( 1\right) f\left( E\left( x\right) \right). \label{hstrEcv1} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>

  </div>
</div> </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">
  <p>We put \(\lambda =1\) in (<a href="#hstrEcv">1</a>) and we obtain (<a href="#hstrEcv1">2</a>). </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="a0000000021">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">11</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If the functions \(f_{i}:A\to \mathbb {R}\), \(i=1,2,\ldots ,k\) are \(h\)-strongly \(E\)-convex on \(A,\) then, for \(a_{i} \geq 0, i=1,2,\ldots ,k\) the function \(F:A\to \mathbb {R}\), \(F\left( x\right) =\overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( x\right) \) is \(h\)-strongly \(E\)-convex on \(A\). </p>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Since the functions \(f_{i}:A\to \mathbb {R}\), \(i=1,2,\ldots ,k\) are \(h\)-strongly \(E\)-convex on \(A,\) then, for each \(x,y\in A\), every \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), we have </p>
<div class="displaymath" id="a0000000023">
  \begin{align*} & F(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\\ & =\overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( \lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) (\alpha y+E\left( y\right) )\right)\\ & \leq h\left( \lambda \right) \overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( E\left( x\right) \right) +h\left( 1-\lambda \right) \overset {k}{\underset {i=1}{\sum }}a_{i}f_{i}\left( E\left( y\right) \right) \\ & =h\left( \lambda \right) F\left( E\left( x\right) \right) +h\left( 1-\lambda \right) F\left( E\left( y\right) \right). \end{align*}
</div>
<p>Hence the function F is \(h\)-strongly \(E\)-convex on \(A\). </p>
<p>We consider a strongly \(E\)-convex set \(A\subset \mathbb {R}^{n},\) a function \(f:\mathbb {R}^{n}\to \mathbb {R}\), and a function \(\varphi :\mathbb {R}\to \mathbb {R}\) linear and nondecreasing. </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="a0000000024">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If the function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is \(h\)-strongly \(E\)-convex on A then the composite function \(\varphi \circ f\) is \(h\)-strongly \(E\)-convex on A. </p>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000025">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Since f is \(h\)-strongly \(E\)-convex on A, for each \(x,y\in A\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\), we have \(f(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\leq h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right)\) and hence </p>
<div class="displaymath" id="a0000000026">
  \begin{align*} & \left( \varphi \circ f\right)(\lambda \left( \alpha x+E\left( x\right) \right) +\left( 1-\lambda \right) \left( \alpha y+E\left( y\right) \right) )\\ & \leq \varphi \left[ h\left( \lambda \right) f\left( E\left( x\right) \right) +h\left( 1-\lambda \right) f\left( E\left( y\right) \right) )\right]\\ & =h\left( \lambda \right) \left( \varphi \circ f\right) \left( E\left( x\right) \right) +h\left(1- \lambda \right) \left( \varphi \circ f\right) \left( E\left( y\right) \right), \end{align*}
</div>
<p> which implies that \(\varphi \circ f\) is \(h\)-strongly \(E\)-convex on A. </p>
<p>We denote \(E(x)\) by \(Ex\) for simplicity. </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="Th">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If the function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is non-negative and differentiable \(h\)-strongly \(E\)-convex on a strongly \(E\)-convex set A and h is a non-negative function with the property \(h\left( \lambda \right) \leq \lambda \) for every \(\lambda \in \left[ 0,1\right] \) then </p>
<div class="equation" id="a0000000027">
<p>
  <div class="equation_content">
    \begin{equation}  \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> for every \(x,y\in A.\) </p>

  </div>
</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">
  <p>Since f is \(h\)-strongly \(E\)-convex on A, </p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  f(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) )\leq h\left( \lambda \right) \left( f\circ E\right) \left( x\right) +h\left( 1-\lambda \right) \left( f\circ E\right) \left( y\right) \end{equation*}
</div>
<p> for each \(x,y\in A\), \(\lambda \in \left[0,1\right] \) and \(\alpha \in \left[0,1\right] \). Since \(h\left( x\right) \leq x\) for every \(x\in \left[ 0,1\right] \) we have </p>
<div class="displaymath" id="a0000000030">
  \begin{align*} & f(\left( \alpha y+Ey\right) +\lambda \left[ \left( \alpha x+Ex\right) -\left( \alpha y+Ey\right) \right] )\\ & \leq \lambda \left( f\circ E\right) \left( x\right) +\left( 1-\lambda \right) \left( f\circ E\right) \left( y\right) \\ & =\left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] \end{align*}
</div>
<p> and hence </p>
<div class="displaymath" id="a0000000031">
  \begin{align*} & f \left( \alpha y+Ey\right) +\lambda \left[ \left( \alpha x+Ex\right) -\left( \alpha y+Ey\right) \right] )\nabla f\left( \alpha y+Ey\right) +O\left( \lambda ^{2}\right) \\ & \leq \left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] \end{align*}
</div>
<p>By taking \(\alpha \rightarrow 0,\) we get </p>
<div class="displaymath" id="a0000000032">
  \begin{align*} & f\left( Ey\right) +\lambda \left( Ex-Ey\right) \nabla f\left( Ey\right) )+O\left( \lambda ^{2}\right)\\ & \leq \left( f\circ E\right) \left( y\right) +\lambda \left[ \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \right] . \end{align*}
</div>
<p>Dividing by \(\lambda {\gt}0\) and taking \(\lambda \rightarrow 0,\) we obtain </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right), \end{equation*}
</div>
<p> for each \(x,y\in A.\) </p>
<p>The following theorem provides a characterization of \(h\)-strongly \(E\)-convex functions with respect to the E-monotonicity of the gradient of map, similar with that obtain from \(E\)-convex functions, by Soleimani-Damaneh in <span class="cite">
	[
	<a href="#Dam11" >3</a>
	]
</span>. </p>
<p><div class="definition_thmwrapper theorem-style-plain" id="a0000000034">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    <sc element at 0x2461613949008>
    </span>
    <span class="definition_thmlabel">14</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(f:\mathbb {R}^{n}\to \mathbb {R}\) be differentiable. The map \(\nabla f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is called E-monotone if</p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0, \end{equation*}
</div>
<p> for every \(x,y\in R^{n}\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="a0000000036">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">15</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If the function \(f:\mathbb {R}^{n}\to \mathbb {R}\) is non-negative and differentiable \(h\)-strongly \(E\)-convex on a strongly \(E\)-convex set A and h is a non-negative function with the property \(h\left( \lambda \right) \leq \lambda \) for every \(\lambda \in \left[ 0,1\right] \) then </p>
<div class="equation" id="a0000000037">
<p>
  <div class="equation_content">
    \begin{equation}  \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0 \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> for every \(x,y\in A.\) </p>

  </div>
</div> </p>
<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">
  <p>Since f is \(h\)-strongly \(E\)-convex on A, from theorem (<a href="#Th">13</a>) we have </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  \left( Ex-Ey\right) \nabla \left( f\circ E\right) \left( y\right) \leq \left( f\circ E\right) \left( x\right) -\left( f\circ E\right) \left( y\right) \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  \left( Ey-Ex\right) \nabla \left( f\circ E\right) \left( x\right) \leq \left( f\circ E\right) \left( y\right) -\left( f\circ E\right) \left( x\right) , \end{equation*}
</div>
<p> for every \(x,y\in A.\) Adding these two inequalities we obtain </p>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  \left( \nabla f(E\left( x\right) )-\nabla f(E\left( y\right) )\right) \left( E\left( x\right) -E\left( y\right) \right) \geq 0 \end{equation*}
</div>
<p> for every \(x,y\in A.\) </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="a0000000042">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    <sc element at 0x2461613961680>
    </span>
    <span class="theorem_thmlabel">16</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let the functions \(g_{i}:\mathbb {R}^{n}\to \mathbb {R}\), \(i=1,2,\ldots ,m\) be \(h\)-strongly \(E\)-convex on \(\mathbb {R}^{n}\). We consider the set </p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  M=\left\{  x\in R^{n}\mid g_{i}\left( x\right) \leq 0,i=1,2,\ldots m\right\} . \end{equation*}
</div>
<p> If \(E\left( M\right) \subseteq M\) and the function h is positively then the set M is strongly \(E\)-convex. </p>

  </div>
</div> </p>
<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">
  <p>Since the functions \(g_{i}:\mathbb {R}^{n}\to \mathbb {R}\), \(i=1,2,\ldots ,m\) are \(h\)-strongly \(E\)-convex on \(\mathbb {R}^{n}\) then, for every \(x,y\in M\), \(\alpha \in \left[ 0,1\right]\) and \(\lambda \in \left[ 0,1\right]\) we have </p>
<div class="displaymath" id="a0000000045">
  \begin{align*} & g_{i}(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) )\\ & \leq h\left( \lambda \right) \left( g_{i}\circ E\right) \left( x\right) +h\left( 1-\lambda \right) \left( g_{i}\circ E\right) \left( y\right) \leq 0, \end{align*}
</div>
<p> and hence \(\lambda \left( \alpha x+Ex\right) +\left( 1-\lambda \right) \left( \alpha y+Ey\right) \in M\). </p>
<p><small class="footnotesize">  </small></p>

  </div>
</div>

  </div>
</div>

  </div>
</div>

  </div>
</div>

  </div>
</div>

  </div>
</div>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
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  <dd><p><i class="sc">Bombardelli, M.</i> and <i class="sc">Varošanec, S.</i>, <i class="it">Properties of \(h\)-convex functions related to the Hermite-Hadamard-Fejér inequalities</i>, Computers and Mathematics with Applications, <b class="bf">58</b>, pp.&#160;1869–1877, 2009. </p>
</dd>
  <dt><a name="Che02">2</a></dt>
  <dd><p><i class="sc">Chen, X.</i>, <i class="it">Some Properties of Semi-\(E\)-convex Functions</i>, J. Math. Anal. Appl, <b class="bf">275</b>, pp.&#160;251–262, 2002. </p>
</dd>
  <dt><a name="Dam11">3</a></dt>
  <dd><p><i class="sc">Soleimani-Damaneh, M.</i>, <i class="it">\(E\)-convexity and its generalizations</i>, International Journal of Computer Mathematics, pp.&#160;1–15, 2011. </p>
</dd>
  <dt><a name="Haz11">4</a></dt>
  <dd><p><i class="sc">Házy, A.</i>, <i class="it">Bernstein-Doetsch Type Results for \(h\)-convex Functions</i>, Mathematical Inequalities and Applications, <b class="bf">14</b>, no.&#160;3, pp.&#160;499–508, 2011. </p>
</dd>
  <dt><a name="EPop72">5</a></dt>
  <dd><p><i class="sc">Popoviciu, E.</i>, <i class="it">Teoreme de medie din analiza matematică şi legătura lor cu teoria interpolării</i>, Editura Dacia, Cluj, 1972. </p>
</dd>
  <dt><a name="TPop45">6</a></dt>
  <dd><p><i class="sc">Popoviciu, T.</i>, <i class="it">Les fonctions convexes</i>, Herman, Paris, 1945. </p>
</dd>
  <dt><a name="Var07">7</a></dt>
  <dd><p><i class="sc">Varošanec, S.</i>, <i class="it">On \(h\)-convexity</i>, J. Math. Anal. Appl, <b class="bf">326</b>, pp.&#160;303–311, 2007. </p>
</dd>
  <dt><a name="You99">8</a></dt>
  <dd><p><i class="sc">Youness, E. A.</i>, <i class="it">\(E\)-convex Sets, \(E\)-convex Functions, and \(E\)-convex Programming</i>, Journal of Optimization Theory and Aplications, <b class="bf">102</b>, no.&#160;2, pp.&#160;439–450, 1999. </p>
</dd>
  <dt><a name="You05a">9</a></dt>
  <dd><p><i class="sc">Youness, E. A.</i> and<i class="sc">Emam, T.</i>, <i class="it">Strongly \(E\)-convex Sets and Strongly \(E\)-convex Functions</i>, Journal of Interdisciplinary Mathematics, <b class="bf">8</b>, no.&#160;1, pp.&#160;107–117, 2005. </p>
</dd>
</dl>


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