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<title>Some applications of quadrature rules for mappings on \(L_p[u,v]\) space <i class="it">via</i> Ostrowski-type inequality: Some applications of quadrature rules for mappings on \(L_p[u,v]\) space <i class="it">via</i> Ostrowski-type inequality</title>
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<h1>Some applications of quadrature rules for mappings on \(L_p[u,v]\) space <i class="it">via</i> Ostrowski-type inequality</h1>
<p class="authors">
<span class="author">Nazia Irshad\(^\ast \) Asif R. Khan\(^\ast \)</span>
</p>
<p class="date">January 6, 2017.</p>
</div>
<p>\(^\ast \)Department of Mathematics, University of Karachi, University Road, Karachi-75270, Pakistan, e-mail: <span class="tt">nazia\(\_ \)irshad@yahoo.com, asifrk@uok.edu.pk</span>. </p>

<div class="abstract"><p> Some Ostrowski-type inequalities are stated for \(L_p [u,v]\) space and for mappings of bounded variations. Applications are also given for obtaining error bounds of some composite quadrature formulae. </p>
<p><b class="bf">MSC.</b> 26D15, 26D20 </p>
<p><b class="bf">Keywords.</b> Ostrowski inequality, \(L_p\) space, bounded variation, numerical integration, sharp bounds. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p> In 1938, Ostrowski introduced a bound for the absolute value of the difference of a function to its average over a finite interval. His well known result named as Ostrowski’s inequality <span class="cite">
	[
	<a href="#Ostbook" >10</a>
	]
</span>. <div class="proposition_thmwrapper " id="a0000000003">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">1</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(g:[u,v]\to \mathbb {R}\) be a differentiable function, also </p>
<div class="displaymath" id="a0000000004">
  \[  |g'(z)|\leq M, \]
</div>
<p> for some positive real constant \(M\), for all \(z\in (u,v)\). Then the following inequality holds for every \(z\in [u,v]\) </p>
<div class="displaymath" id="1.1">
  \begin{eqnarray}  \left|g(z)-\tfrac {1}{v-u}\int _u^v g(s)ds \right|\leq \Big[\tfrac {1}{4}+\tfrac {\left(z-\frac{u+v}{2}\right)^2}{(v-u)^2} \Big](v-u)M,\label{1.1} \end{eqnarray}
</div>
<p> where the constant \(\tfrac {1}{4}\) is sharp. </p>

  </div>
</div> Let \(g, h:[u,v] \rightarrow \mathbb {R}\) be two absolutely continuous functions such that functions and their product are integrable, the Čebyšev functional <span class="cite">
	[
	<a href="#cheb fn" >2</a>
	]
</span> is defined by </p>
<div class="displaymath" id="a0000000005">
  \begin{align*}  T(g,h) & =\tfrac {1}{v-u}\int _u^v\left(g(z) - \tfrac {1}{v-u}\int _u^v g(z) dz \right)\left(h(z) - \tfrac {1}{v-u}\int _u^v h(z) dz \right)dz\\ & =\tfrac {1}{v-u}\int _u^v g(z) h(z) dz -\tfrac {1}{(v-u)^2}\left(\int _u^v g(z) dz \right)\left(\int _u^v h(z) dz \right). \end{align*}
</div>
<p> In 1934, the following result proved by Grüss <span class="cite">
	[
	<a href="#G.gruss" >5</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#mont" >6</a>
	]
</span>): <div class="proposition_thmwrapper " id="a0000000006">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(T(g,h)\) be as defined above, we have </p>
<div class="displaymath" id="1.2">
  \begin{eqnarray}  |T(g,h)|\leq \tfrac {1}{4}(M_1-m_1)(M_2-m_2),\label{1.2} \end{eqnarray}
</div>
<p> where \(m_1,m_2,M_1,M_2\) \(\in \mathbb {R}\) and satisfy the conditions </p>
<div class="displaymath" id="a0000000007">
  \begin{eqnarray*}  m_1\leq g(z)\leq M_1\quad \text{and} \quad m_2\leq h(z)\leq M_2, \quad \forall \;  z\in [u,v]. \end{eqnarray*}
</div>

  </div>
</div> By G.V. Milovanović in <span class="cite">
	[
	<a href="#1st apl" >8</a>
	]
</span>, an application of classical Ostrowski inequality in quadrature formula was given for the very first time, also its generalization to functions in several variables was given in this article. </p>
<p>A generalization of Ostrowski inequality developed by Milovanović and Pečarić <span class="cite">
	[
	<a href="#2nd apl" >9</a>
	]
</span>, which is stated as: <div class="proposition_thmwrapper " id="a0000000008">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(g:\mathbb {R}\to \mathbb {R}\) be \(n({\gt}1)\) times differentiable function such that \(\displaystyle |g^{n}(z)|\leq M\) \((\forall z \in (u,v))\). Then for every \(z \in [u,v]\) </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \begin{equation}  \Big|\tfrac {1}{n}\Big(g(z)+\sum _{k-1}^{n-1}F_k\Big)-\tfrac {1}{v-u}\int _u^v g(s)ds\Big|\leq \tfrac {M}{n(n+1)!}\tfrac {(z-u)^{n+1}+(v-z)^{n+1}}{v-u} \label{1.3} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where \(F_k\) is defined by </p>
<div class="displaymath" id="a0000000009">
  \begin{eqnarray*}  F_k \equiv F_k(g;n;\xi ,u,v)\equiv \tfrac {n-k}{k!}\tfrac {g^{k-1}(u)(z-u)^k-g^{k-1}(v)(z-v)^k}{v-u}. \end{eqnarray*}
</div>

  </div>
</div> In 1997, Dragomir and Wang <span class="cite">
	[
	<a href="#dragW" >3</a>
	]
</span> proved the following proposition by using \((\ref{1.2})\) which is known as Ostrowski-Grüss inequality. <div class="proposition_thmwrapper " id="a0000000010">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">4</span>
  </div>
  <div class="proposition_thmcontent">
  <p>If \(g:[u,v]\to \mathbb {R}\) such that \(g'\) is bounded and </p>
<div class="displaymath" id="a0000000011">
  \[ m\leq g'(z)\leq M, \]
</div>
<p> for all \(z\in [u,v]\) and for real constants \(m\) and \(M\), then </p>
<div class="equation" id="1.4">
<p>
  <div class="equation_content">
    \begin{equation}  \left|g(z) - \tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _u^v g(s)ds\right| \leq \tfrac {1}{4}(v-u)(M-m).\label{1.4} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>

  </div>
</div> </p>
<p>In this paper, we first derive an integral identity for differentiable functions by using the kernel \((\ref{2.1})\). Then, we apply this equality to get our results for functions whose first derivative is bounded. First section is based on introduction and preliminaries. In the second and in the third section we prove inequalities for absolutely continuous mappings in which \(g'\) \(\in \) \(L_p [u,v]\) for \(p\ge 1\) and mappings of bounded variation, respectively. In the last section, we will give some applications for composite quadrature rules. </p>
<h1 id="a0000000012">2 The Case Where \(g\)\(' \in L_{\lowercase {p}} [\lowercase {u},\lowercase {v}]\), \(\lowercase {p}\geq 1 \)</h1>
<p>In order to prove our main results, we need the following lemma from <span class="cite">
	[
	<a href="#drag2" >7</a>
	]
</span>: <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>Let \(g:[u,v]\to \mathbb {R}\) be a function. Then for the kernel \(P(z,s)\) on \([u,v]\) given as </p>
<div class="displaymath" id="2.1">
  \begin{eqnarray}  P(z,s)=\left\{  \begin{array}{rl} s-z+\frac{v-u}{2},& \text{if}\quad s \in [u,z],\\[2mm] s-z-\frac{v-u}{2},& \text{if}\quad s \in (z,v], \end{array} \right.\label{2.1} \end{eqnarray}
</div>
<p> the following identity holds </p>
<div class="displaymath" id="2.2">
  \begin{multline}  \tfrac {1}{v-u}\int _u^v P(z,s)g{’}(s)ds =g(z)-\tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _u^v g(s)ds.\label{2.2} \end{multline}
</div>

  </div>
</div> In this section, we are going to present Ostrowski-type integral inequality for \(g'\in \) \(L_p [u,v]\). <div class="theorem_thmwrapper " id="a0000000014">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(g:I \to \mathbb {R}\) be an absolutely continuous mapping on \(I^o\), the interior of the interval \(I\), where \(u,v \in I\) with \(u{\lt}v\). If \(g{’}\in L_p [u,v]\), for \(p\geq 1\), then we get the following inequality </p>
<div class="displaymath" id="4.1">
  \begin{align} & \left|g(z)-\tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _u^v g(s)ds\right|\leq \label{4.1}\\ & \leq \tfrac {\| g{’}\| _p}{(v-u)(q+1)^{\frac{1}{q}}} \left[\left(\tfrac {u+v}{2}-z \right)^{q+1}+\left(z-\tfrac {u+v}{2} \right)^{q+1}+2\left(\tfrac {v-u}{2}\right)^{q+1} \right]^\frac {1}{q}.\nonumber \end{align}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Using the Hölder inequality in \((\ref{2.2})\), for any \(z\in [u,v]\), we get </p>
<div class="displaymath" id="a0000000016">
  \begin{eqnarray*} & & \left|g(z)-\tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _u^v g(s)ds\right|=\\ & &  =\tfrac {1}{v-u}\left|\int _u^v P(z,s)g’(s)ds\right|\\ & & \leq \tfrac {1}{v-u}\left(\int _u^v|P(z,s)|^qds \right)^\frac {1}{q}\left(\int _u^v|g{’}(s)|^p ds\right)^\frac {1}{p}\\ & & = \tfrac {1}{v-u}\left[\int _u^z \left|s-\left(z-\tfrac {v-u}{2}\right)\right|^q ds + \int _z^v \left|s-\left(z+\tfrac {v-u}{2}\right)\right|^q ds\right]^\frac {1}{q} \| g{’}\| _p\\ & & =\tfrac { \| g{’}\| _p}{(v-u)(q+1)^{\frac{1}{q}}} \left[\left(\tfrac {u+v}{2}-z \right)^{q+1}+\left(z-\tfrac {u+v}{2} \right)^{q+1}+2\left(\tfrac {v-u}{2}\right)^{q+1} \right]^\frac {1}{q}. \end{eqnarray*}
</div>
<p> <div class="proof_wrapper" id="a0000000017">
  <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="remark_thmwrapper " id="a0000000018">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">7</span>
  </div>
  <div class="remark_thmcontent">
  <p>If we substitute \(q=1\) (and \(p=\infty \)) in (<a href="#4.1">7</a>), then we get the following Corollary.<span class="qed">â–¡</span></p>

  </div>
</div> <div class="corollary_thmwrapper " id="a0000000019">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">8</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(g:I \to \mathbb {R}\) be an absolutely continuously mapping on \(I^o\), the interior of the interval I, where \(u, v\) \(\in \) I with \(u{\lt}v\). If \(g{’}\) is bounded on \([u,v]\), then the following inequality holds for any z \(\in \) \([u,v]\) </p>
<div class="equation" id="3.1">
<p>
  <div class="equation_content">
    \begin{equation}  \left|g(z)- \tfrac {g(v)-g(u)}{v-u}\left(z\! -\! \tfrac {u+v}{2}\right)\! -\! \tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \Big[\tfrac {1}{4} + \tfrac {\left(z-\frac{u+v}{2}\right)^2}{(v-u)^2}\Big](v-u) \| g'\| _\infty . \label{3.1} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000020">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">9</span>
  </div>
  <div class="remark_thmcontent">
  <p>The inequality \((\ref{3.1})\) is the generalization of Ostrowski inequality which is presented in Proposition \(5\), \(i.e.\), by replacing \(g(u)=g(v)\) in (<a href="#3.1">8</a>), we get (<a href="#1.3">3</a>) and also by choosing \(\| g'\| _\infty =M\) we get (<a href="#1.1">1</a>).<span class="qed">â–¡</span></p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000021">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">10</span>
  </div>
  <div class="remark_thmcontent">
  <p>If we replace \(z=\tfrac {u+v}{2}\) in (<a href="#3.1">8</a>), then we get the following midpoint inequality </p>
<div class="displaymath" id="3.4">
  \begin{eqnarray}  \left| g\left(\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{4}(v-u) \| g’\| _\infty , \label{3.4} \end{eqnarray}
</div>
<p> where the constant \(\tfrac {1}{4}\) is sharp.<span class="qed">â–¡</span></p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000022">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">11</span>
  </div>
  <div class="remark_thmcontent">
  <p>By replacing \(z=u\) or \(z=v\) in (<a href="#3.1">8</a>), we get the trapezoidal inequality </p>
<div class="equation" id="3.5">
<p>
  <div class="equation_content">
    \begin{equation}  \left|\tfrac {g(u)+g(v)}{2}-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{2}(v-u) \| g'\| _\infty . \label{3.5} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>

  </div>
</div> <div class="corollary_thmwrapper " id="a0000000023">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">12</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(g\) be a function as defined in Theorem \(7\). </p>
<p><span class="rm">1)</span> If we replace \(z=\tfrac {u+v}{2}\) in \((\ref{4.1})\), then we get the midpoint inequality \(\forall \) \(p\geq 1\): </p>
<div class="displaymath" id="4.2">
  \begin{eqnarray}  \left| g\left(\tfrac {u+v}{2}\right) - \tfrac {1}{v-u}\int _u^b g(s)ds \right| \leq \tfrac {1}{2}\left[\tfrac {{(v-u)}}{(q+1)}\right]^\frac {1}{q}\| g’\| _p\label{4.2} \end{eqnarray}
</div>
<p> where the constant \(\tfrac {1}{2}\left[\frac{{(v-u)}}{(q+1)}\right]^\frac {1}{q}\) is sharp. </p>
<p><span class="rm">2)</span> If we replace \(z=u\) or \(z=v\) in \((\ref{4.1})\), we get the trapezoidal inequality \(\forall \) \(p\geq 1\): </p>
<div class="displaymath" id="4.3">
  \begin{multline}  \left|\tfrac {g(u)+g(v)}{2}-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{(v-u)(q+1)^{\frac{1}{q}}}\left[\left(\tfrac {v-u}{2} \right)^{q+1}+3\left(\tfrac {v-u}{2}\right)^{q+1} \right]^\frac {1}{q} \| g’\| _p.\label{4.3} \end{multline}
</div>
<p><span class="rm">a)</span> If \(q\) is odd, then </p>
<div class="displaymath" id="4.4">
  \begin{eqnarray} & & \left|\tfrac {g(u)+g(v)}{2}-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right|\leq \tfrac {1}{2}\left(\tfrac {2(v-u)}{q+1}\right)^\frac {1}{q} \| g’\| _p.\label{4.4} \end{eqnarray}
</div>
<p><span class="rm">b)</span> If \(q\) is even, then </p>
<div class="displaymath" id="4.5">
  \begin{eqnarray} & & \left|\tfrac {g(u)+g(v)}{2}-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right|\leq \tfrac {1}{2}\left(\tfrac {v-u}{q+1}\right)^\frac {1}{q} \| g’\| _p.\label{4.5}\qed \end{eqnarray}
</div>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000024">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">13</span>
  </div>
  <div class="remark_thmcontent">
  <p>By the inequality \((\ref{3.4})\) we retrieve the result of Corollary \(5\) and the inequality \((\ref{4.2})\) gives us the result of Corollary \(8\) of M. W. Alomari paper <span class="cite">
	[
	<a href="#drag1" >1</a>
	]
</span>, respectively.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000025">3 The Case where \(g\) is of Bounded Variation</h1>
<p> <div class="theorem_thmwrapper " id="a0000000026">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(g:[u,v]\to \mathbb {R}\) be a function of bounded variation. Then the following inequality holds for any \(z\) \(\in \) \([u,v]\): </p>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation}  \left|g(z)- \tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)\! -\! \tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{2}\max \left\{ \left|\tfrac {u+v-2z}{v-u}\right|, 1\right\} \bigvee _{u}^{v}g \label{2.3} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> where \(\displaystyle \bigvee _u^v g\) is the total variation of g over \([u,v]\) and the constant \(\frac{1}{2}\) is sharp. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Recalling the definition from <span class="cite">
	[
	<a href="#drag1" >1</a>
	]
</span>, for a continuous function \(p:[c,d]\) \(\to \) \(\mathbb {R}\) and a function \(\nu :[c,d]\) \(\to \) \(\mathbb {R}\) of bounded variation, the following inequality holds: </p>
<div class="displaymath" id="2.4">
  \begin{eqnarray}  \left|\int _c^d p(s)d\nu (s)\right|\leq \sup _{s\in [c,d]} |p(s)| \bigvee _c^d \nu .\label{2.4} \end{eqnarray}
</div>
<p> Now using Lemma \(6\) with the inequality (<a href="#2.4">16</a>) for \(p(s)=P(z,s)\), and \(\nu (s)=g(s)\), \(s\in [u,v]\), we get </p>
<div class="displaymath" id="a0000000028">
  \begin{eqnarray*} & & \left|\tfrac {1}{v-u}\int _u^v P(z,s)d g(s)\right|\leq \\ & & \leq \tfrac {1}{v-u}\left|\int _u^z P(z,s)d g(s)\right|+\tfrac {1}{v-u}\left|\int _z^v P(z,s)d g(s)\right|\\ & &  \leq \tfrac {1}{v-u} \sup _{s\in [u,z]}\left| P(z,s)\right| \bigvee _u^z g+\tfrac {1}{v-u}\sup _{s\in (z,v]}\left| P(z,s)\right| \bigvee _z^v g\\ & &  =\tfrac {1}{v-u}\max \left\{ \left|\tfrac {u+v}{2}-z\right|,\tfrac {v-u}{2}\right\} \bigvee _u^z g +\tfrac {1}{v-u}\max \left\{ \left|\tfrac {u+v}{2}-z\right|\right\} \bigvee _z^v g:=M(z).\end{eqnarray*}
</div>
<p> We notice that </p>
<div class="displaymath" id="a0000000029">
  \begin{eqnarray*}  M(z)\leq \tfrac {1}{v-u} \max \left\{  \left|\tfrac {u+v}{2}-z\right|,\tfrac {v-u}{2}\right\}  \Big[\bigvee _u^z g +\bigvee _z^v g\Big] =\tfrac {1}{2} \max \left\{ \left|\tfrac {u+v-2z}{v-u}\right|, 1\right\} \bigvee _u^v g \end{eqnarray*}
</div>
<p> which proves the inequality (<a href="#2.3">15</a>). </p>
<p>To prove that the constant \(\tfrac {1}{2}\) in inequality (<a href="#2.3">15</a>) is sharp, we suppose that the inequality (<a href="#2.3">15</a>) is valid for a constant \(K{\gt}0\), <i class="it">i.e.</i>, </p>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.5} \left|g(z)- \tfrac {g(v)-g(u)}{v-u}\left(z-\tfrac {u+v}{2}\right)\! -\! \tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \!  \leq K \max \left\{ \left|\tfrac {u+v-2z}{v-u}\right|, 1\right\}  \bigvee _u^v g \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> for any \(z\) \(\in \) \([u,v]\). </p>
<p>Consider the mapping \(g:[u,v]\to \{ 0,1\} \) is defined as </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  g(s)=\left\{  \begin{array}{rl} 0,& \quad s \in (u,v)\\ 1,& \quad s \in \{ u,v\} . \end{array} \right. \end{eqnarray*}
</div>
<p> For \(z=u\), we have </p>
<div class="displaymath" id="a0000000031">
  \begin{eqnarray*}  \int _u^v g(s)ds=0 \quad {\rm and} \quad \bigvee _u^v g=2. \end{eqnarray*}
</div>
<p> By using \((\ref{2.5})\), we obtain, </p>
<div class="displaymath" id="a0000000032">
  \[ 1\leq 2K\  \text{or}\  \tfrac {1}{2}\leq K, \]
</div>
<p> and thus it is proved that the constant \(\tfrac {1}{2}\) is sharp. <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> <div class="corollary_thmwrapper " id="a0000000034">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">15</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(g\) be a function as defined in Theorem \(15\). </p>
<p><span class="rm">1)</span> If we replace \(z=\frac{u+v}{2}\) in \((\ref{2.3})\), then we get the midpoint inequality </p>
<div class="displaymath" id="2.6">
  \begin{eqnarray}  \left| g\left(\tfrac {u+v}{2}\right)-\tfrac {1}{v-u}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{2}\bigvee _{u}^{v} g \label{2.6} \end{eqnarray}
</div>
<p> where the constant \(\tfrac {1}{2}\) is sharp. </p>
<p><span class="rm">2)</span> If we replace \(z=u\) or \(z=v\) in \((\ref{2.3})\), then we get the trapezoidal inequality </p>
<div class="displaymath" id="2.7">
  \begin{eqnarray}  \left|\tfrac {g(u)+g(v)}{2} -\tfrac {1}{u-v}\int _{u}^{v}g(s)ds\right| \leq \tfrac {1}{2}\bigvee _{u}^{v} g \label{2.7} \end{eqnarray}
</div>
<p> where the constant \(\frac{1}{2}\) is sharp. </p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000035">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">16</span>
  </div>
  <div class="remark_thmcontent">
  <p>The inequalities \((\ref{2.6})\) and \((\ref{2.7})\) are the results of Corollary \(2\) of M.W. Alomari paper <span class="cite">
	[
	<a href="#drag1" >1</a>
	]
</span> and the Corollaries \(2.6\) and \(2.4\) of S.S. Dragomir paper <span class="cite">
	[
	<a href="#drag1a" >4</a>
	]
</span>, respectively.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000036">4 Applications to Numerical Quadrature Rules</h1>
<p> Now, we are going to discuss some applications in numerical quadrature rules, which can be used to get some sharp bounds. </p>
<p>Let \(I_n:u=z_0{\lt}z_1{\lt}\cdots {\lt}z_n=v\) be a partition of the interval \([u,v]\) and let \(\Delta z_k=z_{k+1}-z_k, k\in \{ 0,1,2,\cdots ,n-1\} .\) Then </p>
<div class="displaymath" id="5.1">
  \begin{eqnarray}  \sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds=Q_n(I_n,g)+R_n(I_n,g)\label{5.1} \end{eqnarray}
</div>
<p> Consider a general quadrature formula </p>
<div class="displaymath" id="5.2">
  \begin{eqnarray}  Q_n(I_n,g):=\sum _{k=0}^{n-1} \left[g(\xi _k)-\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k-\tfrac {z_{k+1}+z_k}{2}\right)\right]\label{5.2} \end{eqnarray}
</div>
<p> for all \(\xi _k \in [a,b].\) <div class="theorem_thmwrapper " id="a0000000037">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">17</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let g be defined as in Theorem \(7\). Then \((\ref{5.1})\) holds where \(Q_n(I_n,g)\) is given by formula \((\ref{5.2})\) and the remainder \(R_n (I_n, g)\) satisfies the estimates </p>
<div class="displaymath" id="5.5">
  \begin{align} & \left|R_n(I_n,g)\right|\leq \label{5.5}\\ & \leq \tfrac {1}{\Delta z_k}\left(\tfrac {1}{q+1}\right)^\frac {1}{q} \sum _{k=0}^{n-1}\left[\left(\tfrac {z_k+z_{k+1}}{2}-\xi _k\right)^{q+1}\! \! \! +\! \left(\xi _k-\tfrac {z_k+z_{k+1}}{2}\right)^{q+1}\! \! \! +\! 2\left(\tfrac {\Delta z_k}{2}\right)^{q+1} \right]^\frac {1}{q}\| g’\| _p\nonumber \end{align}
</div>
<p> for all \(\xi _k \in [z_k, z_{k+1}].\) </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>Applying inequality \((\ref{4.1})\) on the intervals, \([z_k, z_{k+1}]\), we can state that </p>
<div class="displaymath" id="a0000000039">
  \begin{eqnarray*}  R_k(I_k,g)=\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds-g(\xi _k) -\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k - \tfrac {z_k+z_{k+1}}{2}\right) \end{eqnarray*}
</div>
<p> we sum the inequalities presented above over \(k\) from \(0\) to \(n-1\). This gives </p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  R_n(I_n,g) =\sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(t)dt-\sum _{k=0}^{n-1}\left[g(\xi _k) -\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k - \tfrac {z_k+z_{k+1}}{2}\right)\right]. \end{equation*}
</div>
<p> It follows from \((\ref{4.1})\) that </p>
<div class="displaymath" id="a0000000041">
  \begin{align*} & \left|R_n(I_n,g)\right|=\\ & =\left|\sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds-\sum _{k=0}^{n-1}\left[g(\xi _k) -\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k - \tfrac {z_k+z_{k+1}}{2}\right)\right]\right|\\ & \leq \tfrac {1}{\Delta z_k}\left(\tfrac {1}{q+1}\right)^\frac {1}{q}\sum _{k=0}^{n-1}\left[\left(\tfrac {z_k+z_{k+1}}{2}-\xi _k\right)^{q+1} \! \! \! +\! \left(\xi _k-\tfrac {z_k+z_{k+1}}{2}\right)^{q+1}\right. \! \! \! +\! \left.2\left(\tfrac {\Delta z_k}{2}\right)^{q+1} \right]^\frac {1}{q}\! \! \| g’\| _p. \end{align*}
</div>
<p> <div class="corollary_thmwrapper " id="a0000000042">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">18</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(q=1\) \((and \quad p=\infty )\) in \((\ref{5.5})\). Then \((\ref{5.1})\) holds, where \(Q_n(I_n,g)\) is given by formula \((\ref{5.2})\) and the remainder \(R_n (I_n, g)\) satisfies the estimate </p>
<div class="displaymath" id="5.4">
  \begin{eqnarray}  \left|R_n(I_n,g)\right| \leq \sum _{k=0}^{n-1}\Delta z_k\bigg[\tfrac {1}{4}+\tfrac {\big(\xi _k- \frac{z_k+z_{k+1}}{2}\big)^2}{(\Delta z_k)^2}\bigg]\| g’\| _\infty \label{5.4} \end{eqnarray}
</div>
<p> for all \(\xi _k \in [z_k, z_{k+1}].\) </p>

  </div>
</div> <div class="theorem_thmwrapper " id="a0000000043">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">19</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(g\) be a function as defined in Theorem \(15\). Then \((\ref{5.1})\) holds, where \(Q_n(I_n,g)\) is given by formula \((\ref{5.2})\) and the remainder satisfies the estimate </p>
<div class="displaymath" id="5.3">
  \begin{eqnarray}  \left|R_n(I_n,g)\right| \leq \displaystyle \sum _{k=0}^{n-1}\tfrac {1}{2}\max \left\{ \left|\tfrac {z_k+z_{k+1}-2\xi _k}{z_{k+1}-z_k}\right|,1\right\} \bigvee _{z_k}^{z_{k+1}}g\label{5.3}\end{eqnarray}
</div>
<p> for all \(\xi _k \in [z_k, z_{k+1}].\) </p>

  </div>
</div> <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>Applying inequality \((\ref{2.3})\) on the intervals, \([z_k, z_{k+1}]\), we can state that </p>
<div class="displaymath" id="a0000000045">
  \begin{eqnarray*}  R_k(I_k,g)=\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds-g(\xi _k) -\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\big(\xi _k - \tfrac {z_k+z_{k+1}}{2}\big). \end{eqnarray*}
</div>
<p> We sum the inequalities presented above over \(k\) from \(0\) to \(n-1\). This gives </p>
<div class="displaymath" id="a0000000046">
  \begin{multline*}  R_n(I_n,g) =\sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds-\sum _{k=0}^{n-1}\left[g(\xi _k) +\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k - \tfrac {z_k+z_{k+1}}{2}\right)\right]. \end{multline*}
</div>
<p> It follows from \((\ref{2.3})\) that </p>
<div class="displaymath" id="a0000000047">
  \begin{align*}  \left|R_n(I_n,g)\right|& =\left|\sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\int _{z_k}^{z_{k+1}}g(s)ds\! -\! \sum _{k=0}^{n-1}\left[g(\xi _k) +\tfrac {g(z_{k+1})-g(z_k)}{\Delta z_k}\left(\xi _k \! -\!  \tfrac {z_k+z_{k+1}}{2}\right)\right]\right|\\ & \leq \sum _{k=0}^{n-1}\tfrac {1}{2}\max \left\{ \left|\tfrac {z_k+z_{k+1}-2\xi _k}{\Delta z_k}\right|, 1\right\} \bigvee _{z_k}^{z_{k+1}}g. \qquad \qed \end{align*}
</div>
<p>If we choose, </p>
<div class="displaymath" id="a0000000048">
  \[ \xi _k=\tfrac {z_k+z_{k+1}}{2} \]
</div>
<p> in \((\ref{5.2})\), then quadrature formula becomes: </p>
<div class="displaymath" id="5.6">
  \begin{eqnarray}  Q_n(I_n,g):=\sum _{k=0}^{n-1} \left[g\left(\tfrac {z_k+z_{k+1}}{2}\right)\right].\label{5.6} \end{eqnarray}
</div>
<p> <div class="remark_thmwrapper " id="a0000000049">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">20</span>
  </div>
  <div class="remark_thmcontent">
  <p>If (<a href="#5.1">20</a>) holds and \(Q_n(I_n,g)\) is given by formula (<a href="#5.6">25</a>). </p>
<ol class="enumerate">
  <li><p>Let \(g\) be as in Theorem \(7\) where the remainder \(R_n(I_n,g)\) from (<a href="#4.2">11</a>) and (<a href="#3.4">9</a>) becomes respectively </p>
<div class="displaymath" id="a0000000050">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\left(\tfrac {1}{q+1}\right)^\frac {1}{q}\sum _{k=0}^{n-1}\left[\Delta z_k\right]^\frac {1}{q}\| g{’}\| _p \end{eqnarray*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000051">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{4}\sum _{k=0}^{n-1}\left[{\Delta z_k}\right]\| g’\| _\infty . \end{eqnarray*}
</div>
</li>
  <li><p>Let \(g\) be as in Theorem \(15\) where the remainder \(R_n(I_n,g)\) from (<a href="#2.6">18</a>) becomes </p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\sum _{k=0}^{n-1}\bigvee _{z_k}^{z_{k+1}}g. \end{equation*}
</div>
</li>
</ol>

  </div>
</div> </p>
<p>If we choose, \(\xi _k=z_k\) or \(\xi _k=z_{k+1}\) in \((\ref{5.2})\), then quadrature formula becomes: </p>
<div class="displaymath" id="5.7">
  \begin{eqnarray}  Q_n(I_n,g):=\sum _{k=0}^{n-1} \left[\tfrac {g(z_k)+g(z_{k+1})}{2}\right].\label{5.7} \end{eqnarray}
</div>
<p> <div class="remark_thmwrapper " id="a0000000053">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">21</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \((\ref{5.1})\) holds and \(Q_n(I_n,g)\) is given by formula \((\ref{5.7})\). </p>
<ol class="enumerate">
  <li><p>Let \(g\) be as in Theorem \(7\) where the remainder \(R_n(I_n,g)\) from \((\ref{4.3})\), \((\ref{4.4})\), \((\ref{4.5})\) and \((\ref{3.5})\) becomes respectively </p>
<div class="displaymath" id="a0000000054">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{(q+1)^{\frac{1}{q}}}\sum _{k=0}^{n-1}\tfrac {1}{\Delta z_k}\left[\left(\tfrac {-\Delta z_k}{2} \right)^{q+1}+3\left(\tfrac {\Delta z_k}{2}\right)^{q+1} \right]^\frac {1}{q} \| g{’}\| _p, \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000055">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\left(\tfrac {2}{q+1}\right)^\frac {1}{q}\sum _{k=0}^{n-1} (\Delta z_k)^\frac {1}{q} \| g{’}\| _p, \end{eqnarray*}
</div>
<div class="displaymath" id="a0000000056">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\left(\tfrac {1}{q+1}\right)^\frac {1}{q}\sum _{k=0}^{n-1}(\Delta z_k)^\frac {1}{q} \| g{’}\| _p \end{eqnarray*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000057">
  \begin{eqnarray*}  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\sum _{k=0}^{n-1}\left[{\Delta z_k}\right]\| g’\| _\infty . \end{eqnarray*}
</div>
</li>
  <li><p>Let \(g\) be as in Theorem \(15\) where the remainder \(R_n(I_n,g)\) from \((\ref{2.7})\) becomes </p>
<div class="displaymath" id="a0000000058">
  \[  \left|R_n(I_n,g)\right| \leq \tfrac {1}{2}\sum _{k=0}^{n-1}\bigvee _{z_k}^{z_{k+1}}g.  \]
</div>
</li>
</ol>

  </div>
</div> </p>
<h1 id="a0000000059">5 Conclusion</h1>
<p> We have given some remarks on Ostrowski type inequalities for absolutely continuous functions in which \(g'\in L_p\) space. Using the results of \(L_p\) space, we have also given some special results for \(L_\infty \) space. Our Corollary \(9\) of Theorem \(7\) is the generalization of Ostrowski inequality<span class="cite">
	[
	<a href="#Ostbook" >10</a>
	]
</span> which is presented in 1938 by A. M Ostrowski. Furthermore, by putting suitable substitutions we get midpoint and trapezoidal rules which are presented in <span class="cite">
	[
	<a href="#drag1" >1</a>
	, 
	<a href="#drag1a" >4</a>
	]
</span>. At the end we have also given some applications for numerical integration. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="drag1">1</a></dt>
  <dd><p><a href ="DOI: 10.1007/s11253-012-0661-x"> <i class="sc">M. W. Alomari</i>, <i class="it">A Companion of Dragomir’s Generalization of the Ostrowski inequality and Applications to Numerical integration</i>, Ukrainian Mathematical Journal, <b class="bf">64</b> (4), (2012), pp.&#160;491–509. <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="cheb fn">2</a></dt>
  <dd><p><i class="sc">P. L. Chebyshev</i>, <i class="it">Sur les expressions approximative des integrals par les auters prises entre les mmes limites</i>, Proc. Math. Soc. Charko\(\upsilon \) , <b class="bf">2</b> (1882), pp.&#160;93–98. </p>
</dd>
  <dt><a name="dragW">3</a></dt>
  <dd><p><a href ="DOI: 10.1007/s11253-012-0661-x"> <i class="sc">S. S. Dragomir</i> and <i class="sc">S.Wang</i>, <i class="it">An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules</i>, Comput. Math. Appl., <b class="bf">33</b> (1997), pp.&#160;15–20. <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="drag1a">4</a></dt>
  <dd><p><i class="sc">S. S. Dragomir</i>, <i class="it">A Companion of Ostrowski’s Inequality for functions of Bounded Variation and Applications</i>, Int. J. Nonlinear Anal. Appl., <b class="bf">5</b> (2014), pp.&#160;89–97. </p>
</dd>
  <dt><a name="G.gruss">5</a></dt>
  <dd><p><a href =" DOI: 10.1007/BF01201355"> <i class="sc">G. Grüss</i>, <i class="it">Uber das Maximum des absoluten Betrages von \(\frac{1}{(b-a)}\int _a^bf(x)g(x)dx-\frac{1}{(b-a)^2}\) \(\times \int _a^bf(x)dx\int _a^bg(x)dx\)</i>, Math. Z., <b class="bf">39</b> (1935), pp.&#160;215–226. <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="mont">6</a></dt>
  <dd><p><i class="sc">Asif R. Khan</i>, <i class="sc">Josip Pečarić</i> and <i class="sc">M. Praljak</i>, <i class="it">Weighted Montgomery inequalities for higher order differentiable functions of two variables</i>, Rev. Anal. Numer. Theor. Approx., <b class="bf">42</b> (1) (2013), pp.&#160;49–71. </p>
</dd>
  <dt><a name="drag2">7</a></dt>
  <dd><p><a href ="DOI: 10.1142/S0219530513500309"> <i class="sc">Mohammas Masjed-Jamei</i> and <i class="sc">Severs. Dragomir</i>, <i class="it">Generalization of Ostrowski-Grüss Inequality</i>, World Scientific, <b class="bf">12</b> (2) (2014), pp.&#160;117–130. <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="1st apl">8</a></dt>
  <dd><p><i class="sc">G. V. Milovanović</i>, <i class="it">On some integral inequalities</i>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 498–541 (1975), pp.&#160;119–124. </p>
</dd>
  <dt><a name="2nd apl">9</a></dt>
  <dd><p><i class="sc">G.V. Milovanović</i>, <i class="sc">J. E. Pečarić</i>, <i class="it">On generalization of the inequality of A. Ostrowski and some related applications</i>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 544–576 (1976), pp.&#160;155–158. </p>
</dd>
  <dt><a name="Ostbook">10</a></dt>
  <dd><p><a href ="DOI: 10.1007/BF01214290"> <i class="sc">A. M. Ostrowski</i>, <i class="it">\(\ddot{U}\)ber die absolutabweichung einer differentiebaren funktion von ihren integralmittelwert</i>, Comment. Math. Helv., <b class="bf">10</b> (1938), pp.&#160;226–227. <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="measureBook">11</a></dt>
  <dd><p><i class="sc">Richard L. Wheeden</i> and <i class="sc">Antoni Zygmund</i>, <i class="it">Measure and Integral</i>, Marcel Dekker. Inc, New York, 1977. </p>
</dd>
</dl>


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