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<h1>Schur convexity properties of the weighted arithmetic integral mean and Chebyshev functional\(^\ast \)</h1>
<p class="authors">
<span class="author">Long Bo-Yong\(^\S \), Jiang Yue-Ping\(^\bullet \) Chu Yu-Ming\(^\S \)</span>
</p>
<p class="date">August 10, 2012.</p>
</div>
<p>\(^\S \)School of Mathematics Science, Anhui University, Feixi Str. no.3, 230039, Hefei, China, e-mail: <span class="tt">lbymxy@163.com</span>. </p>
<p>\(^\bullet \)College of Mathematics and Econometrics, Hunan University, Lushan nan Str. no.2, 410082, Changsha, China, e-mail: <span class="tt">ypjiang@hnu.edu.cn</span>. </p>
<p>\(^\S \)Department of Mathematics, Huzhou Teachers College, Xueshi Str. no.1, 313000, Huzhou, China, e-mail: <span class="tt">chuyuming2005@aliyun.com</span>. </p>
<p>\(^*\)The work of this author has been supported by the Natural Science Foundation of China (Grant No. 11071069), and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924) </p>

<div class="abstract"><p> In this paper, we discuss the Schur convexity, Schur geometrical convexity and Schur harmonic convexity of the weighted arithmetic integral mean and Chebyshev functional. Several sufficient conditions, and necessary and sufficient conditions are established. </p>
<p><b class="bf">MSC.</b> 26B25, 05E05. </p>
<p><b class="bf">Keywords.</b> Weighted arithmetic integral mean, Weighted Chebyshev functional, Schur convex, Schur geometrically convex, Schur harmonic convex. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>   Throughout this paper, we use \(\mathbb {R}^{n}\) to denote the \(n\)-dimensional Euclidean space \((n\geq 2)\) and \(\mathbb {R}_{+}^{n}=\{ (x_{1},x_{2},\cdots ,x_{n}): x_{i}{\gt}0,i=1,2,\cdots ,n\} \). In particular, we use \(\mathbb {R}\) to denote \(\mathbb {R}^{1}\). </p>
<p>For the sake of convenience, we use the following notation system. </p>
<p>For \(x=(x_{1},x_{2},\cdots ,x_{n})\), \(y=(y_{1},y_{2},\cdots ,y_{n})\in \mathbb {R}_{+}^{n}\) and \(\alpha \in \mathbb {R}_{+}\), we denote by </p>
<div class="displaymath" id="a0000000003">
  \begin{align*}  x\pm y& =(x_{1}\pm y_{1},x_{2}\pm y_{2},\cdots ,x_{n}\pm y_{n}),\\ xy& =(x_{1}y_{1},x_{2}y_{2},\cdots ,x_{n}y_{n}),\\ \alpha x& =(\alpha x_{1},\alpha x_{2},\cdots ,\alpha x_{n}),\\ x^{\alpha }& =(x_{1}^{\alpha },x_{2}^{\alpha },\cdots ,x_{n}^{\alpha }),\\ \log x& =(\log x_{1},\log x_{2},\cdots ,\log x_{n}) \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \tfrac {1}{x}=\left(\tfrac {1}{x_{1}},\tfrac {1}{x_{2}},\cdots ,\tfrac {1}{x_{n}}\right). \end{equation*}
</div>
<p><div class="definition_thmwrapper " id="a0000000005">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p>A real-valued function \(F\) on \(E\subseteq \mathbb {R}^{n}\) is said to be Schur convex if </p>
<div class="equation" id="a0000000006">
<p>
  <div class="equation_content">
    \begin{equation}  F(x_{1},x_{2},\cdots ,x_{n})\leq F(y_{1},y_{2},\cdots ,y_{n})\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> for each pair of \(n\)-tuples \(x=(x_{1},\cdots ,x_{n})\), \(y=(y_{1},\cdots ,y_{n}) \in E\), such that \(x\prec y\), i.e. </p>
<div class="displaymath" id="a0000000007">
  \[ \sum _{i=1}^{k}x_{[i]}\leq \sum _{i=1}^{k}y_{[i]},\quad k=1,2,\cdots ,n-1 \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000008">
  \[ \sum _{i=1}^{n}x_{[i]}=\sum _{i=1}^{n}y_{[i]}, \]
</div>
<p> where \(x_{[i]}\) denotes the \(i\)th largest component of \(x\). \(F\) is called Schur concave if \(-F\) is Schur convex. </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000009">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
  </div>
  <div class="definition_thmcontent">
  <p>A real-valued function \(F\) on \(E\subseteq \mathbb {R}_{+}^{n}\) is said to be Schur geometrically convex if \({\rm (1)}\) holds for each pair of \(n\)-tuples \(x=(x_{1},x_{2},\cdots ,x_{n})\), \(y=(y_{1},y_{2},\cdots ,y_{n})\in E\), such that \(\log x\prec \log y\). \(F\) is called Schur geometrically concave if \(\tfrac {1}{F}\) is Schur geometrically convex. </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000010">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
  </div>
  <div class="definition_thmcontent">
  <p>A real-valued function \(F\) on \(E\subseteq \mathbb {R}_{+}^{n}\) is said to be Schur harmonic convex if \({\rm (1)}\) holds for each pair of \(n\)-tuples \(x=(x_{1},x_{2},\cdots ,x_{n})\), \(y=(y_{1},y_{2},\cdots ,y_{n}) \in E\), such that \(\tfrac {1}{x}\prec \tfrac {1}{y}\). \(F\) is called Schur harmonic concave on \(E\) if inequality <span class="rm">(1)</span> is reversed. </p>

  </div>
</div> </p>
<p>The Schur convexity was introduced by I. Schur [1] in 1923 and it has many important applications in analytic inequalities [2-7], extended mean values [8-11] and other related fields. Recently, the Schur geometrical and harmonic convexities were investigated in [4, 5, 12, 13, 14]. </p>
<p><div class="lemma_thmwrapper " id="a0000000011">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="rm">(see [2])</span>. Let \(E\subseteq \mathbb {R}^{n}\) be a symmetric convex set with nonempty interior \({\rm int}E\) and \(f:E\to \mathbb {R}\) be a continuous symmetric function. If \(f\) is differentiable on \({\rm int}E\), then \(f\) is Schur convex on \(E\) if and only if </p>
<div class="equation" id="a0000000012">
<p>
  <div class="equation_content">
    \begin{equation}  (x_{1}-x_{2})\left(\tfrac {\partial f}{\partial x_{1}}-\tfrac {\partial f}{\partial x_{2}}\right)\geq 0 \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> for all \(x=(x_{1},\cdots ,x_{n})\in {\rm int}E\). \(f\) is Schur concave on \(E\) if and only if inequality <span class="rm">(2)</span> is reversed. Here \(E\) is a symmetric set means that \(x\in E\) implies \(Px\in E\) for any \(n\times n\) permutation matrix \(P\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000013">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="rm">(see [13])</span>. Let \(E\subseteq \mathbb {R}_{+}^{n}\) be a symmetric geometrically convex set with nonempty interior \({\rm int}E\) and \(f:E\to \mathbb {R}_{+}\) be a continuous symmetric function. If \(f\) is differentiable on \({\rm int}E\), then \(f\) is Schur geometrically convex on \(E\) if and only if </p>
<div class="displaymath" id="a0000000014">
  \[ (\log x_{1}-\log x_{2})\left(x_{1}\tfrac {\partial f}{\partial x_{1}}-x_{2}\tfrac {\partial f}{\partial x_{2}}\right)\geq 0 \]
</div>
<p> for all \(x=(x_{1},x_{2},\cdots ,x_{n})\in intE\). Here \(E\) is a geometrically convex set means that \(x^{\tfrac {1}{2}}y^{\tfrac {1}{2}}\in E\) whenever \(x,y\in E\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000015">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="rm">(see [14])</span>. Let \(E\subseteq \mathbb {R}_{+}^{n}\) be a symmetric harmonic convex set with nonempty interior \({\rm int}E\) and \(f:E\to \mathbb {R}_{+}\) be a continuous symmetric function. If \(f\) is differentiable on \({\rm int}E\), then \(f\) is Schur harmonic convex on \(E\) if and only if </p>
<div class="displaymath" id="a0000000016">
  \[  (x_{1}-x_{2})\left(x_{1}^{2}\tfrac {\partial f}{\partial x_{1}}-x_{2}^{2}\tfrac {\partial f}{\partial x_{1}}\right)\geq 0 \]
</div>
<p> for all \(x=(x_{1},x_{2},\cdots ,x_{n})\in {\rm int}E\). Here \(E\) is a harmonic convex set means that \(\tfrac {2xy}{x+y}\in E\) whenever \(x,y\in E\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000017">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">7</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(f\) be a continuous function on \(I\subseteq \mathbb {R}\) and \(p\) be a positive continuous weight on \(I\). Then the well-known weighted arithmetic integral mean \(F_{p}(x,y)\) is defined by </p>
<div class="equation" id="a0000000018">
<p>
  <div class="equation_content">
    \begin{equation} \qquad \qquad F_{p}(x,y)=\left\{  \begin{array}{llll}\tfrac {1}{\int ^{y}_{x}p(t){\rm d}t}\displaystyle {\int ^{y}_{x}p(t)f(t){\rm d}t}, &  x,y\in I,x\neq y,\\ f(x), &  x=y. \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>If \(p(t)=1\), then <span class="rm">(3)</span> reduces to the arithmetic integral mean </p>
<div class="equation" id="a0000000019">
<p>
  <div class="equation_content">
    \begin{equation} \qquad \qquad F(x,y)=\left\{  \begin{array}{llll}\tfrac {1}{y-x}\displaystyle {\int ^{y}_{x}f(t){\rm d}t}, &  x,y\in I,x\neq y,\\ f(x), &  x=y. \end{array}\right.\hspace{2.cm} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>

  </div>
</div> </p>
<p>Recently, the weighted arithmetic integral mean \(F_{p}(x,y)\) has been the subject of intensive research [8, 15]. In particular, the following Theorems 8-10 can be found in the literature [16-18]. </p>
<p><div class="theorem_thmwrapper " id="a0000000020">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\) be a continuous function on \(I\), let \(p\) be a positive continuous weight on \(I\). Then the weighted arithmetic integral mean \(F_{p}(x,y)\) is Schur convex (concave) on \(I^{2}\) if and only if the inequality </p>
<div class="displaymath" id="a0000000021">
  \begin{eqnarray}  \tfrac {1}{\int ^{y}_{x}p(t){\rm d}t}\int ^{y}_{x}p(t)f(t){\rm d}t\leq \tfrac {p(x)f(x)+p(y)f(y)}{p(x)+p(y)} \end{eqnarray}
</div>
<p>holds (reverses) for all \(x,y\in I\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000022">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\) be a continuous function on \(I\). Then the arithmetic integral mean \(F(x,y)\) defined as in <span class="rm">(4)</span> is Schur convex (concave) on \(I^{2}\) if and only if \(f\) is convex (concave) on \(I\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000023">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\) be a second order differentiable function on \(I\) with \(3f'(x)+xf''(x)\geq 0\)(\(\leq 0\)). Then the arithmetic integral mean \(F(x,y)\) is Schur geometrically convex (concave) on \(I^{2}\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000024">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">11</span>
  </div>
  <div class="definition_thmcontent">
  <p>The weighted Chebyshev functional \(T(f,g,p)\) is defined by </p>
<div class="displaymath" id="a0000000025">
  \begin{eqnarray}  T(f,g,p)& =& T(f,g,p;x,y)\\ & =& \tfrac {\int ^{y}_{x}p(t)f(t)g(t){\rm d}t}{\int ^{y}_{x}p(t){\rm d}t}-\tfrac {\int ^{y}_{x}p(t)f(t){\rm d}t}{\int ^{y}_{x}p(t){\rm d}t} \cdot \tfrac {\int ^{y}_{x}p(t)g(t){\rm d}t}{\int ^{y}_{x}p(t){\rm d}t},\nonumber \end{eqnarray}
</div>
<p> where \(p(t)\) is a positive Lebesgue measurable function on \([x,y]\) such that \(0{\lt}\int ^{y}_{x}p(t){\rm d}t{\lt}+\infty \), \(f\), \(g: [x,y]\rightarrow \mathbb {R}\) are two Lebesgue measurable functions on \([x,y]\) and the integrals in <span class="rm">(6)</span> are assumed to exist. </p>
<p>If \(p(t)=1\), then <span class="rm">(6)</span> becomes the Chebyshev functional </p>
<div class="equation" id="a0000000026">
<p>
  <div class="equation_content">
    \begin{equation}  T(f,g)=T(f,g;x,y)=\tfrac {\displaystyle {\int ^{y}_{x}f(t)g(t){\rm d}t}}{y-x}-\tfrac {\displaystyle {\int ^{y}_{x}f(t){\rm d}t}}{y-x} \cdot \tfrac {\displaystyle {\int ^{y}_{x}g(t){\rm d}t}}{y-x}. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>

  </div>
</div> </p>
<p>The weighted Chebyshev functional has a long history and an extensive repertoire of applications in many fields in including numerical quadrature, transform theory, probability and statistical problems, and special functions [19-24]. C̆uljak [25] proved that </p>
<p><div class="theorem_thmwrapper " id="a0000000027">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f\) and \(g\) are monotonic in the same sense (in the opposite sense) on \(I\), then the Chebyshev functional \(T(f,g)\) is Schur convex (concave) on ÔÚ\(I^{2}\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000028">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(p\) be a positive continuous weight on \(I\). Then the weighted Chebyshev functional \(T(f,g,p)\) is Schur convex (concave) on \(I^{2}\) if and only if the inequality </p>
<div class="displaymath" id="a0000000029">
  \begin{align*} & T(f,g,p;x,y)\leq \tfrac {p(x)\big(f(x)-\overline{f_{p}}(x,y)\big)\big(g(x)-\overline{g_{p}}(x,y)\big)+p(y)\big(f(y)-\overline{f_{p}}(x,y)\big)\big(g(y)-\overline{g_{p}}(x,y)\big)}{p(x)+p(y)} \end{align*}
</div>
<p> holds (reverses) for all \(x,y\in I\). Here, </p>
<div class="displaymath" id="a0000000030">
  \begin{align}  P(x,y)& =\int ^{y}_{x}p(t){\rm d}t,\nonumber \\ \overline{f_{p}}& =\overline{f_{p}}(x,y)=\tfrac {1}{P(x,y)}\int ^{y}_{x}p(t)f(t){\rm d}t,\nonumber \\ \overline{g_{p}}& =\overline{g_{p}}(x,y)=\tfrac {1}{P(x,y)}\int ^{y}_{x}p(t)g(t){\rm d}t. \end{align}
</div>

  </div>
</div> </p>
<p>Our main purpose of this paper is to discuss the Schur convexity, Schur geometrical convexity and Schur harmonic convexity of the weighted arithmetic integral mean and Chebyshev functional. </p>
<h1 id="a0000000031">2 Schur convexity of the weighted arithmetic integral mean</h1>
<p> <div class="theorem_thmwrapper " id="a0000000032">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\) be a continuous function on \(I\) and \(p\) be a positive continuous weight on \(I\). Then the weighted arithmetic integral mean \(F_{p}(x,y)\) defined by <span class="rm">(3)</span> on \(I^{2}\) is <span class="rm">(1)</span> Schur geometrically convex if and only if </p>
<div class="displaymath" id="a0000000033">
  \begin{eqnarray}  \overline{f_{p}}(x,y)\leq \tfrac {xp(x)f(x)+yp(y)f(y)}{xp(x)+yp(y)} \end{eqnarray}
</div>
<p>for all \(x,y\in I\); <span class="rm">(2)</span> Schur harmonic convex if and only if </p>
<div class="displaymath" id="a0000000034">
  \begin{eqnarray}  \overline{f_{p}}(x,y)\leq \tfrac {x^{2}p(x)f(x)+y^{2}p(y)f(y)}{x^{2}p(x)+y^{2}p(y)} \end{eqnarray}
</div>
<p>for all \(x,y\in I\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000035">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>From (3) and (8) one has </p>
<div class="displaymath" id="a0000000036">
  \[ \tfrac {\partial F_{p}(x,y)}{\partial x}=\tfrac {-p(x)f(x)P(x,y)+p(x)\int ^{y}_{x}p(t)f(t){\rm d}t}{P^{2}(x,y)}, \]
</div>
<div class="displaymath" id="a0000000037">
  \[ \tfrac {\partial F_{p}(x,y)}{\partial y}=\tfrac {-p(y)f(y)P(x,y)+p(y)\int ^{y}_{x}p(t)f(t){\rm d}t}{P^{2}(x,y)}. \]
</div>
<p>(1) Simple computations lead to </p>
<div class="displaymath" id="a0000000038">
  \begin{align*} & (\log y-\log x)\left(y\tfrac {\partial F_{p}(x,y)}{\partial y}-x\tfrac {\partial F_{p}(x,y)}{\partial x}\right)=\\ & =\tfrac {\log y-\log x}{P(x,y)}\left[xp(x)f(x)+yp(y)f(y)-xp(x)\overline{f_{p}}(x,y)-yp(y)\overline{f_{p}}(x,y)\right]. \end{align*}
</div>
<p>We clearly see that \(\tfrac {\log y-\log x}{P(x,y)}\geq 0\), then by Lemma 5 we know that \(F_{p}(x,y)\) is Schur geometrically convex if and only if </p>
<div class="displaymath" id="a0000000039">
  \[ xp(x)f(x)+yp(y)f(y)-xp(x)\overline{f_{p}}(x,y)-yp(y)\overline{f_{p}}(x,y)\geq 0 \]
</div>
<p> for all \(x,y\in I\). That is to say for all \(x,y\in I\), </p>
<div class="displaymath" id="a0000000040">
  \[ \overline{f_{p}}(x,y)\leq \tfrac {xp(x)f(x)+yp(y)f(y)}{xp(x)+yp(y)}. \]
</div>
<p>(2) Direct computations yield </p>
<div class="displaymath" id="a0000000041">
  \begin{eqnarray*} & & (y-x)\left(y^{2}\tfrac {\partial F_{p}(x,y)}{\partial y}-x^{2}\tfrac {\partial F_{p}(x,y)}{\partial x}\right)=\\ & & =\tfrac {y-x}{P(x,y)}[x^{2}p(x)f(x)+y^{2}p(y)f(y)-x^{2}p(x)\overline{f_{p}}(x,y)-y^{2}p(y)\overline{f_{p}}(x,y)]. \end{eqnarray*}
</div>
<p>Since \(\tfrac { y- x}{P(x,y)}\geq 0\), by Lemma 6 we know that \(F_{p}(x,y)\) is Schur harmonic convex if and only if for all \(x,y\in I\) it holds </p>
<div class="displaymath" id="a0000000042">
  \[ x^{2}p(x)f(x)+y^{2}p(y)f(y)-x^{2}p(x)\overline{f_{p}}(x,y)-y^{2}p(y)\overline{f_{p}}(x,y)\geq 0, \]
</div>
<p> i.e. </p>
<div class="displaymath" id="a0000000043">
  \[ \overline{f_{p}}(x,y)\leq \tfrac {x^{2}p(x)f(x)+y^{2}p(y)f(y)}{x^{2}p(x)+y^{2}p(y)} \]
</div>
<p> for all \(x,y\in I\). </p>
<p>Let </p>
<div class="equation" id="a0000000044">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{f}=\overline{f}(x,y)=\tfrac {1}{y-x}\int ^{y}_{x}f(t){\rm d}t,\quad \overline{g}=\overline{g}(x,y)=\tfrac {1}{y-x}\int ^{y}_{x}g(t){\rm d}t, \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> and take \(p(t)=1\). Then Theorem 14 leads to the following Corollary 15. </p>
<p><div class="corollary_thmwrapper " id="a0000000045">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">15</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(f\) be a continuous function on \(I\), then the arithmetic integral mean \(F(x,y)\) on \(I^{2}\) is </p>
<p><span class="rm">(1)</span> Schur geometrically convex if and only if </p>
<div class="equation" id="a0000000046">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{f}(x,y)\leq \tfrac {xf(x)+yf(y)}{x+y} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> for all \(x,y\in I\); </p>
<p><span class="rm">(2)</span> Schur harmonic convex if and only if </p>
<div class="equation" id="a0000000047">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{f}(x,y)\leq \tfrac {x^{2}f(x)+y^{2}f(y)}{x^{2}+y^{2}} \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> for all \(x,y\in I\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000048">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">16</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(p\) be a positive continuous weight on \(I\), \(f\) be a differentiable function on \(I\) with \(f'(y)\geq \tfrac {p(y)}{P(x,y)}\cdot \tfrac {f(y)-f(x)}{y-x}\) for any \(x,y\in I\), \(F_{p}(x,y)\) be the weighted arithmetic integral mean. Then the following statements are true. </p>
<p><span class="rm">(1)</span> If \(f\) and \(p\) have the same monotonicity on \(I\), then \(F_{p}(x,y)\) is Schur convex on \(I^{2}\); </p>
<p><span class="rm">(2)</span> If \(f(t)\) and \(tp(t)\) have the same monotonicity on \(I\), then \(F_{p}(x,y)\) is Schur geometrically convex on \(I^{2}\); </p>
<p><span class="rm">(3)</span> If \(f(t)\) and \(t^{2}p(t)\) have the same monotonicity on \(I\), then \(F_{p}(x,y)\) is Schur harmonic convex on \(I^{2}\). </p>

  </div>
</div> </p>
<p><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>For any \(x,y\in I\), let </p>
<div class="displaymath" id="a0000000050">
  \[ G(x,y)=2\int ^{y}_{x}p(t)f(t){\rm d}t-[f(x)+f(y)]\int ^{y}_{x}p(t){\rm d}t. \]
</div>
<p> Then </p>
<div class="equation" id="a0000000051">
<p>
  <div class="equation_content">
    \begin{equation}  G(x,x)=0, \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<div class="equation" id="a0000000052">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\partial G(x,y)}{\partial y}=p(y)[f(y)-f(x)]-f'(y)\int ^{y}_{x}p(t){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> If </p>
<div class="equation" id="a0000000053">
<p>
  <div class="equation_content">
    \begin{equation}  f'(y)\geq \tfrac {p(y)}{P(x,y)}\cdot \tfrac {f(y)-f(x)}{y-x}, \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> then equation (15) and inequality (16) lead to the conclusion that </p>
<div class="equation" id="a0000000054">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {\partial G(x,y)}{\partial y}\leq 0. \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> Equation (14) and inequality (17) imply </p>
<div class="equation" id="a0000000055">
<p>
  <div class="equation_content">
    \begin{equation}  G(x,y)\leq 0. \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> Inequality (18) leads to </p>
<div class="equation" id="a0000000056">
<p>
  <div class="equation_content">
    \begin{equation}  \overline{f_{p}}(x,y)\leq \tfrac {f(x)+f(y)}{2}. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>(1) If \(f\) and \(p\) have the same monotonicity on \(I\), then </p>
<div class="displaymath" id="a0000000057">
  \[ (f(y)-f(x))[p(y)-p(x)]\geq 0 \]
</div>
<p> for any \(x,y\in I\). It follows that </p>
<div class="equation" id="a0000000058">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {f(x)+f(y)}{2}\leq \tfrac {p(x)f(x)+p(y)f(y)}{p(x)+p(y)}. \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>From inequalities (19) and (20) together with Theorem 8 we clearly see that \(F_{p}(x,y)\) is Schur convex on \(I^{2}\). </p>
<p>(2) If \(f(t)\) and \(tp(t)\) have the same monotonicity on \(I\), then </p>
<div class="displaymath" id="a0000000059">
  \[ (f(y)-f(x))[yp(y)-xp(x)]\geq 0 \]
</div>
<p> for any \(x,y\in I\). So, we have </p>
<div class="equation" id="a0000000060">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {f(x)+f(y)}{2}\leq \tfrac {xp(x)f(x)+yp(y)f(y)}{xp(x)+yp(y)}. \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>From inequalities (19) and (21) together with Theorem 14(1) we known that \(F_{p}(x,y)\) is Schur geometrically convex on \(I^{2}\). </p>
<p>(3) If \(f(t)\) and \(t^{2}p(t)\) have the same monotonicity on \(I\), then </p>
<div class="displaymath" id="a0000000061">
  \[ (f(y)-f(x))[y^{2}p(y)-x^{2}p(x)]\geq 0 \]
</div>
<p> for any \(x,y\in I\). Hence, we get </p>
<div class="equation" id="a0000000062">
<p>
  <div class="equation_content">
    \begin{equation} \tfrac {f(x)+f(y)}{2}\leq \tfrac {x^{2}p(x)f(x)+y^{2}p(y)f(y)}{x^{2}p(x)+y^{2}p(y)} \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>for any \(x,y\in I\). </p>
<p>It follows from inequalities (19) and (22) together with Theorem 14(2) that \(F_{p}(x,y)\) is Schur harmonic convex on \(I^{2}\). </p>
<p><div class="theorem_thmwrapper " id="a0000000063">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">17</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\) be a continuous, increasing (decreasing) and convex (concave) function on \(I\). Then the arithmetic integral mean \(F(x,y)\) is Schur geometrically and harmonic convex (concave) on \(I^{2}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000064">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>If \(f\) is convex on \(I\), then by the well known Hermite-Hadamard inequality we have </p>
<div class="displaymath" id="a0000000065">
  \begin{eqnarray}  \overline{f}(x,y)=\tfrac {1}{y-x}\int ^{y}_{x}f(t){\rm d}t\leq \tfrac {f(x)+f(y)}{2}. \end{eqnarray}
</div>
<p> If \(f\) is increasing on \(I\), then </p>
<div class="displaymath" id="a0000000066">
  \[ (f(y)-f(x))(y-x)\geq 0 \]
</div>
<p> for any \(x,y\in I\). Therefore, </p>
<div class="equation" id="a0000000067">
<p>
  <div class="equation_content">
    \begin{equation} \tfrac {f(x)+f(y)}{2}\leq \tfrac {xf(x)+yf(y)}{x+y}. \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>From inequalities (23) and (24) together with Corollary 15(1) we clearly see that \(F(x,y)\) is Schur geometrically convex on \(I^{2}\). </p>
<p>If \(f\) is increasing on \(I\), then </p>
<div class="displaymath" id="a0000000068">
  \[ (f(y)-f(x))(y^{2}-x^{2})\geq 0 \]
</div>
<p>for any \(x,y\in I\). Hence, </p>
<div class="equation" id="a0000000069">
<p>
  <div class="equation_content">
    \begin{equation}  \tfrac {f(x)+f(y)}{2}\leq \tfrac {x^{2}f(x)+y^{2}f(y)}{x^{2}+y^{2}}. \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>From inequalities (23) and (25) together with Corollary 15(2) we known that \(F(x,y)\) is Schur harmonic convex on \(I^{2}\). </p>
<h1 id="a0000000070">3 Schur convexity of the weighted Chebyshev functional</h1>
<p> It is not difficult to verify that the weighted Chebyshev functional \(T(f,g,p)\) satisfy the identity: </p>
<div class="equation" id="a0000000071">
<p>
  <div class="equation_content">
    \begin{equation}  T(f,g,p;x,y)=\tfrac {1}{\int ^{y}_{x}p(t){\rm d}t}\int ^{y}_{x}p(t)(f(t)-\overline{f_{p}}(x,y))(g(t)-\overline{g_{p}}(x,y)){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p>For fixed \(x\) and \(y\) with \(x\neq y\), let </p>
<div class="equation" id="a0000000072">
<p>
  <div class="equation_content">
    \begin{equation}  G_{p}(t)=(f(t)-\overline{f_{p}}(x,y))(g(t)-\overline{g_{p}}(x,y)) \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> for all \(t\in [x,y]\). Then (26) can be rewritten as </p>
<div class="equation" id="a0000000073">
<p>
  <div class="equation_content">
    \begin{equation}  T(f,g,p,x,y)=\tfrac {1}{\int ^{y}_{x}p(t)dt}\int ^{y}_{x}p(t)G_{p}(t){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p>Equation (28) shows that the weighted Chebyshev functional \(T(f,g,p)\) can be expressed by the weighted arithmetic integral mean of \(G_{p}(t)\). </p>
<p>If \(p(t)=1\), then (27) and (28) can be rewritten as </p>
<div class="equation" id="a0000000074">
<p>
  <div class="equation_content">
    \begin{equation}  G(t)=(f(t)-\overline{f}(x,y))(g(t)-\overline{g}(x,y)), \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<div class="equation" id="a0000000075">
<p>
  <div class="equation_content">
    \begin{equation}  T(f,g)=T(f,g; x,y)=\tfrac {1}{y-x}\int ^{y}_{x}G(t){\rm d}t, \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> respectively. From (27), (28) and Theorem 14, we have </p>
<p><div class="theorem_thmwrapper " id="a0000000076">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">18</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(p\) be a positive continuous weight on \(I\). Then the weighted Chebyshev functional \(T(f,g,p)\) defined as in <span class="rm">(6)</span> is </p>
<p><span class="rm">(1)</span>&#8195;Schur geometrically convex on \(I^{2}\) if and only if the inequality </p>
<div class="displaymath" id="a0000000077">
  \begin{eqnarray*}  T(f,g,p;x,y)\leq \tfrac {x p(x)(f(x)-\overline{f_{p}}(x,y))(g(x)-\overline{g_{p}}(x,y))+y p(y)(f(y)-\overline{f_{p}}(x,y))(g(y)-\overline{g_{p}}(x,y))}{x p(x)+y p(y)} \end{eqnarray*}
</div>
<p> holds for all \(x,y\in I\); </p>
<p><span class="rm">(2)</span>&#8195;Schur harmonic convex on \(I^{2}\) if and only if the inequality </p>
<div class="displaymath" id="a0000000078">
  \begin{eqnarray*} T(f,g,p;x,y)\leq \tfrac {x^{2}p(x)(f(x)-\overline{f_{p}}(x,y))(g(x)-\overline{g_{p}}(x,y))+y^{2}p(y)(f(y)-\overline{f_{p}}(x,y))(g(y)-\overline{g_{p}}(x,y))}{x^{2}p(x)+y^{2}p(y)} \end{eqnarray*}
</div>
<p>holds for all \(x,y\in I\). </p>

  </div>
</div> </p>
<p>From Theorems 13 and 18, we get the following Corollary 19. </p>
<p><div class="corollary_thmwrapper " id="a0000000079">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">19</span>
  </div>
  <div class="corollary_thmcontent">
  <p>The Chebyshev functional \(T(f,g)\) defined as in <span class="rm">(7)</span> is </p>
<p><span class="rm">(1)</span> Schur convex on \(I^{2}\) if and only if </p>
<div class="displaymath" id="a0000000080">
  \begin{eqnarray*}  T(f,g;x,y)\leq \tfrac {(f(x)-\overline{f}(x,y))(g(x)-\overline{g}(x,y))+(f(y)-\overline{f}(x,y))(g(y)-\overline{g}(x,y))}{2} \end{eqnarray*}
</div>
<p> for all \(x,y\in I\); </p>
<p><span class="rm">(2)</span> Schur geometrically convex on \(I^{2}\) if and only if </p>
<div class="displaymath" id="a0000000081">
  \begin{eqnarray*}  T(f,g;x,y)\leq \tfrac {x(f(x)-\overline{f}(x,y))(g(x)-\overline{g}(x,y))+y(f(y)-\overline{f}(x,y))(g(y)-\overline{g}(x,y))}{x+y} \end{eqnarray*}
</div>
<p> for all \(x,y\in I\); </p>
<p><span class="rm">(3)</span> Schur harmonic convex on \(I^{2}\) if and only if </p>
<div class="displaymath" id="a0000000082">
  \begin{eqnarray*} T(f,g;x,y)\leq \tfrac {x^{2}(f(x)-\overline{f}(x,y))(g(x)-\overline{g}(x,y))+y^{2}(f(y)-\overline{f}(x,y))(g(y)-\overline{g}(x,y))}{x^{2}+y^{2}} \end{eqnarray*}
</div>
<p>for all \(x,y\in I\). </p>

  </div>
</div> </p>
<p>From (28) and Theorem 16, we have the following Theorem 20. </p>
<p><div class="theorem_thmwrapper " id="a0000000083">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">20</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(p\) be a positive continuous weight on \(I\), \(G_{p}(t)=(f(t)-\overline{f_{p}}(x,y))(g(t)-\overline{g_{p}}(x,y))\) be a differentiable function on \(I\) with \(G_{p}'(y)\geq \tfrac {p(y)}{P(x,y)}\cdot \tfrac {G(y)-G(x)}{y-x}\) for any \(x,y\in I\). Then the following statements are true. </p>
<p><span class="rm">(1)</span> If \(G_{p}(t)\) and \(p(t)\) have the same monotonicity on \(I\), then \(T(f,g,p)\) is Schur convex on \(I^{2}\). </p>
<p><span class="rm">(2)</span> If \(G_{p}(t)\) and \(tp(t)\) have the same monotonicity on \(I\), then \(T(f,g,p)\) is Schur geometrically convex on \(I^{2}\). </p>
<p><span class="rm">(3)</span> If \(G_{p}(t)\) and \(t^{2}p(t)\) have the same monotonicity on \(I\), then \(T(f,g,p)\) is Schur harmonic convex on \(I^{2}\). </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="a0000000084">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">21</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(G(t)=(f(t)-\overline{f}(x,y))(g(t)-\overline{g}(x,y))\) be a continuous function on \([x,y]\subseteq I\) for any \(x,y\in I\). Then the following statements are true. </p>
<p><span class="rm">(1)</span>&#8195;\(T(f,g)\) is Schur convex (concave) on \(I^{2}\) if and only if \(G(t)\) is convex (concave) on \(I\). </p>
<p><span class="rm">(2)</span>&#8195;If \(G(t)\) is increasing (decreasing) and convex (concave) on \(I\), then \(T(f,g)\) is Schur geometrically and harmonic convex (concave) on \(I^{2}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000085">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Theorem 21(1) follows from (30) and Theorem 9, and Theorem 21(2) follows form (30) and Theorem 17. </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000086">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</dd>
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  <dd><p><i class="sc">S. S. Dragomir</i> and <i class="sc">Th. M. Rassias</i>, <i class="it">Ostrowski type inequalities and applicatons in numerical integration</i>, Kluwer Academic, Dordrecht, 2002. </p>
</dd>
  <dt><a name="20">20</a></dt>
  <dd><p><i class="sc">P. Cerone</i> and <i class="sc">S.S. Dragomir</i>, <i class="it">New bounds for the C̆ebyšev functional</i>, Appl. Math. Lett., <b class="bf">18</b>(6) (2005), pp. 603–611. </p>
</dd>
  <dt><a name="21">21</a></dt>
  <dd><p><i class="sc">P. Cerone</i>, <i class="it">On Chebyshev funtional bounds</i>, Differential Difference Equations and Applications, Hindawi Publ. Corp., New York, 2006, pp. 267–277. </p>
</dd>
  <dt><a name="22">22</a></dt>
  <dd><p><i class="sc">S. S. Dragomir</i>, <i class="it">Some integral inequalities of Grüss type</i>, Indian J. Pure Appl. Math., <b class="bf">31</b>(4) (2002), pp. 397–415. </p>
</dd>
  <dt><a name="23">23</a></dt>
  <dd><p><i class="sc">N. Elezović, Lj. Marangunić</i> and <i class="sc">J. Pec̆arić</i>, <i class="it">Some improvements of Grüss type inequality</i>, J. Math. Inequal., <b class="bf">1</b>(3) (2007), pp. 425–436. </p>
</dd>
  <dt><a name="24">24</a></dt>
  <dd><p><i class="sc">F. Zafar</i> and <i class="sc">N. A. Mir</i>, <i class="it">A note on the generalization of some new Čebyšev type inequlities</i>, Tamsui Oxf. J. Inf. Math. Sci., <b class="bf">27</b>(2) (2011), pp. 149–157. </p>
</dd>
  <dt><a name="25">25</a></dt>
  <dd><p><i class="sc">V. C̆uljak</i>, <i class="it">Schur-convexity of the weighted C̆ebyšev functional</i>, J. Math. Inequal., <b class="bf">5</b>(2) (2011), pp. 213–217. </p>
</dd>
</dl>


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