<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Double inequalities for quadrature formula of Gauss Type: Double inequalities for quadrature formula of Gauss Type</title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">


<div class="titlepage">
<h1>Double inequalities for quadrature formula of Gauss Type</h1>
<p class="authors">
<span class="author">Marius Heljiu\(^\S \)</span>
</p>
<p class="date">September 13, 2011</p>
</div>
<p>\(^\S \)Department of Mathematics, University of Petroşani, Romania,<br />e-mail: <span class="tt">mheljiu@upet.ro</span>. </p>

<div class="abstract"><p> Double inequalities for the remainder term of the Gauss quadrature formula are given. These inequalities are sharp. It also will consider particular cases for n = 1, 2. </p>
<p><b class="bf">MSC.</b> 26D10, 65D30 </p>
<p><b class="bf">Keywords.</b> Quadrature rule, double integral inequalities, numerical integration. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>In this work we will consider Gauss’s quadrature rule(see <span class="cite">
	[
	<a href="#2" >2</a>
	]
</span>) </p>
<div class="equation" id="1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{1} \int _a^b f(x){\rm d}x= \sum _{i=1}^{n} C_if(x_i)+\  R[f] \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p> where the remainder term for \(f:[a,b]\to \mathbb {R}\), \(f\in C^{2n}[a,b]\), has the representation </p>
<div class="equation" id="2">
<p>
  <div class="equation_content">
    \begin{equation}  \label{2} R[f]=\int _a^b \varphi (x)f^{(2n)}(x){\rm d}x. \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p>The nodes \(x_i\), \(i=\overline{1,n}\) from quadrature formula (1.1) are given by the relation </p>
<div class="equation" id="3">
<p>
  <div class="equation_content">
    \begin{equation}  \label{3} x_i=\displaystyle \tfrac {a+b}{2}+\displaystyle \tfrac {b-a}{2}\xi _i \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p> where \(\xi _i \) is replaced by the roots \(\xi _1, \xi _2,..., \xi _n\) of Legendre polynomial </p>
<div class="equation" id="4">
<p>
  <div class="equation_content">
    \begin{equation}  \label{4} X_n(\xi )=\displaystyle \tfrac {1}{2^n n!}\displaystyle \tfrac {d^n(1-\xi ^2)^n}{d\xi ^n}. \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p> and the coefficients \(C_i\), \(i=\overline{1,n}\), from Gauss’s formula shall be determined by putting the conditions that the quadrature formula must be accurate for any polynomial of degree \(2n-1\). </p>
<p>The function \(\varphi (x)\), from remaining term expression, being symmetric to the line \(x=\tfrac {a+b}{2}\), is sufficient to provide the equation \(y=\varphi (x)\) in the intervals \([a,x_1], [x_1,x_2],[x_2,x_3]...\) to middle of interval (a,b). So the function \(\varphi (x)\) coincide on all the the intervals \([a,x_1], [x_1,x_2],[x_2,x_3]...,\) with the relations: </p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{l} \varphi _1(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s\\ \varphi _2(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s-C_1\tfrac {(x-x_1)^{2n-1}}{(2n-1)!}\\ \varphi _3(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s-C_1\tfrac {(x-x_1)^{2n-1}}{(2n-1)!}-C_2\tfrac {(x-x_2)^{2n-1}}{(2n-1)!}\\ \cdots \end{array} \end{equation}
  </div>
  <span class="equation_label">1.5</span>
</p>
</div>
<p>In the next paragraph we will establish double integral inequalities for the remainder term of the Gauss quadrature formula. We will also establish conditions under which the results are sharp. At the end of the paper we will analyze the particular cases of Gauss formula with one respectively two nodes. </p>
<h1 id="a0000000004">2 Main results</h1>
<p> In this section we prove the following theorems: <div class="theorem_thmwrapper " id="t1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(f\in C^{2n}[a,b]\) then </p>
<div class="displaymath" id="5">
  \begin{align}  \label{5} &  M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ &  \le \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x\leq \nonumber \\ &  \le M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}
</div>
<p> where \(\gamma ,\Gamma \in \mathbb {R}\), \(\gamma \le f^{(2n)}(x)\le \Gamma \), for all \(x\in [a,b]\) and \(S_{2n-1}=\displaystyle \tfrac {f^{(2n-1)}(b)-f^{(2n-1)}(a)}{b-a}\). Moreover, </p>
<div class="equation" id="6">
<p>
  <div class="equation_content">
    \begin{equation}  \label{6} \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \end{equation}
  </div>
  <span class="equation_label">2.2</span>
</p>
</div>
<p> the inequalities <span class="rm">(5)</span> are sharp. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000005">
  <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 (<a href="#1">1.1</a>) and (<a href="#2">1.2</a>), we obtain </p>
<div class="equation" id="7">
<p>
  <div class="equation_content">
    \begin{equation}  \label{7} \int _a^b \varphi (x)f^{(2n)}(x){\rm d}x=\int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i). \end{equation}
  </div>
  <span class="equation_label">2.3</span>
</p>
</div>
<p> But it is well-known from [3] pp 283 that </p>
<div class="equation" id="8">
<p>
  <div class="equation_content">
    \begin{equation}  \label{8} \int _a^b \varphi (x){\rm d}x=\tfrac {(n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}. \end{equation}
  </div>
  <span class="equation_label">2.4</span>
</p>
</div>
<p>From (<a href="#7">2.3</a>) and (<a href="#8">2.4</a>) it follows </p>
<div class="displaymath" id="9">
  \begin{align}  \label{9} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x &  =\int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i)\\ & \quad -\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}
</div>
<p> and </p>
<div class="displaymath" id="10">
  \begin{align}  \label{10} \int _a^b [\Gamma -f^{(2n)}(x)]\varphi (x){\rm d}x &  =-\int _a^b f(x){\rm d}x+\sum _{i=1}^{n}C_if(x_i)\\ & \quad +\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}
</div>
<p>On the other hand, we have </p>
<div class="equation" id="11">
<p>
  <div class="equation_content">
    \begin{equation}  \label{11} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x\le \max _{x\in [a,b]}|\varphi (x)|\int _a^b |f^{(2n)}(x)-\gamma |{\rm d}x. \end{equation}
  </div>
  <span class="equation_label">2.7</span>
</p>
</div>
<p> By calculating the right member of the inequality(<a href="#11">2.7</a>), we get </p>
<div class="displaymath" id="12">
  \begin{align}  \label{12} \int _a^b |f^{(2n)}(x)-\gamma |{\rm d}x &  =\int _a^b (f^{(2n)}(x)-\gamma ){\rm d}x\\ &  =f^{(2n-1)}(b)-f^{(2n-1)}(a)-\gamma (b-a)\nonumber \\ &  =(S_{2n-1}-\gamma )(b-a).\nonumber \end{align}
</div>
<p> From (<a href="#11">2.7</a>), (<a href="#12">2.8</a>) and noting </p>
<div class="equation" id="13">
<p>
  <div class="equation_content">
    \begin{equation}  \label{13} \max _{x\in [a,b]}|\varphi (x)|=M, \end{equation}
  </div>
  <span class="equation_label">2.9</span>
</p>
</div>
<p> we obtain </p>
<div class="displaymath" id="14">
  \begin{align}  \label{14} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x &  \leq M(S_{2n-1}-\gamma )(b-a). \end{align}
</div>
<p> From (<a href="#9">2.5</a>), by using previous relation, result </p>
<div class="equation" id="144">
<p>
  <div class="equation_content">
    \begin{equation}  \label{144} \int _a^b f(x){\rm d}x-\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ \le \displaystyle M(S_{2n-1}-\gamma )(b-a)+\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)} \end{equation}
  </div>
  <span class="equation_label">2.11</span>
</p>
</div>
<p>In the same way we have </p>
<div class="equation" id="15">
<p>
  <div class="equation_content">
    \begin{equation}  \label{15} \int _a^b [\Gamma -f^{(2n)}(x)]\varphi (x){\rm d}x \le \max _{x\in [a,b]}|\varphi (x)|\int _a^b |\Gamma -f^{(2n)}(x)|{\rm d}x \end{equation}
  </div>
  <span class="equation_label">2.12</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="16">
  \begin{align}  \label{16} \int _a^b |\Gamma -f^{(2n)}(x)|{\rm d}x &  =\int _a^b (\Gamma -f^{(2n)}(x)){\rm d}x\\ &  =\Gamma (b-a)-f^{(2n-1)}(b)+f^{(2n-1)}(a)\nonumber \\ &  =(\Gamma -S_{2n-1})(b-a)\nonumber \end{align}
</div>
<p> Using (<a href="#10">2.6</a>), (<a href="#13">2.9</a>), (<a href="#15">2.12</a>) and (<a href="#16">2.13</a>) we obtain the inequality </p>
<div class="displaymath" id="17">
  \begin{align}  \label{17} &  -\int _a^b f(x){\rm d}x+\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ &  \leq M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}.\nonumber \end{align}
</div>
<p> The inequalities (<a href="#5">2.1</a>) follow from the inequalities (<a href="#144">2.11</a>) and (<a href="#17">2.14</a>). </p>
<p>To prove the second part of the theorem we consider the function \(f(x)=(x-a)^{2n}\). I have \( f^{(2n)}(x)=(2n)!, \gamma =\Gamma =(2n)!\) and \(S_{2n-1}=(2n)!\). It is easy to show that all the three members of the double inequality (<a href="#5">2.1</a>) are equal. This completes the proof. <div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The following theorem offer us \(R[f]\) and analogous inequality with the one given by Theorem <a href="#t1">2.1</a>. <div class="theorem_thmwrapper " id="t2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">2.1</span>, we have </p>
<div class="displaymath" id="18">
  \begin{align}  \label{18} &  M(\gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ &  \le \int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i)\leq \nonumber \\ &  \le M(\Gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}
</div>
<p> Moreover, </p>
<div class="displaymath" id="a0000000007">
  \[ \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \]
</div>
<p> the inequalities <span class="rm">(18)</span> are sharp. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By using the relations (<a href="#7">2.3</a>), (<a href="#9">2.5</a>), (<a href="#11">2.7</a>) and (<a href="#12">2.8</a>) it follows </p>
<div class="displaymath" id="19">
  \begin{align}  \label{19} &  -\int _a^b f(x){\rm d}x+\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ &  \le -M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}
</div>
<p>Analogous by using the relations (<a href="#7">2.3</a>), (<a href="#10">2.6</a>), (<a href="#15">2.12</a>) and (<a href="#16">2.13</a>) we obtain </p>
<div class="displaymath" id="20">
  \begin{align}  \label{20} &  \int _a^b f(x){\rm d}x-\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ &  \le M(\Gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}
</div>
<p>From the relations (<a href="#19">2.16</a>) and (<a href="#20">2.17</a>) result the inequalities (<a href="#18">2.15</a>). To prove that the double inequalities (<a href="#18">2.15</a>) are exact we follow the steps of the proof from Theorem&#160;<a href="#t1">2.1</a>. <div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Theorem <a href="#t3">2.3</a> gives us the inequalities which do not depend on \(S_3\). <div class="theorem_thmwrapper " id="t3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">1</span>, we have </p>
<div class="displaymath" id="21">
  \begin{align}  \label{21} &  \displaystyle \tfrac {1}{2}M(\gamma -\Gamma )(b-a)-\displaystyle \tfrac {\gamma +\Gamma }{2}\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}\\ &  \le \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x\nonumber \\ &  \le \displaystyle \tfrac {1}{2}M(\Gamma -\gamma )(b-a)-\displaystyle \tfrac {\gamma +\Gamma }{2}\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}.\nonumber \end{align}
</div>
<p>If </p>
<div class="displaymath" id="a0000000010">
  \[ \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \]
</div>
<p> then the inequalities <span class="rm">(21)</span> are sharp. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Multiplying the inequality (<a href="#18">2.15</a>) with \((-1)\) we obtain </p>
<div class="displaymath" id="22">
  \begin{align}  \label{22} &  -M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ &  \le -\int _a^b f(x){\rm d}x+\sum _{i=1}^{n}C_if(x_i)\leq \nonumber \\ &  \le -M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}
</div>
<p> From the inequalities (<a href="#5">2.1</a>) and (<a href="#22">2.19</a>) we have </p>
<div class="displaymath" id="23">
  \begin{align}  \label{23} &  \displaystyle M(\gamma -\Gamma )(b-a)-\displaystyle (\gamma +\Gamma )\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}\\ &  \le 2\left[ \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x \right]\nonumber \\ &  \le \displaystyle M(\Gamma -\gamma )(b-a)-\displaystyle (\gamma +\Gamma )\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}.\nonumber \end{align}
</div>
<p>Multiplying (<a href="#23">2.20</a>) with \(\tfrac {1}{2}\) results the double inequality (<a href="#21">2.18</a>). Considering the function \(f(x)=(x-a)^{2n}\) we will show that the inequality is sharp. This completes the proof. <div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000013">3 Particular cases</h1>
<p>Using the results of the before paragraph, we will present the following double inequalities for the rest of Gauss quadrature formulas in the cases n=1, n=2. </p>
<p>The Gauss’s formula with a single node (n = 1), also called the mid-point, has the form </p>
<div class="equation" id="a0000000014">
<p>
  <div class="equation_content">
    \begin{equation}  \int _a^b f(x){\rm d}x=(b-a)\left[f\left(\tfrac {a+b}{2}\right)\right] +R[f], \end{equation}
  </div>
  <span class="equation_label">3.1</span>
</p>
</div>
<p> where </p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation}  R[f]=\int _a^b \varphi (x)f''(x){\rm d}x, \end{equation}
  </div>
  <span class="equation_label">3.2</span>
</p>
</div>
<p> The function \(\varphi (x)\) is given by the relation </p>
<div class="equation" id="24">
<p>
  <div class="equation_content">
    \begin{equation}  \label{24} \varphi (x) =\left\{ \begin{array}{lll} \displaystyle \tfrac {(x-a)^2}{2}, &  \mbox{if} &  x\in \left[a,\displaystyle \tfrac {a+b}{2}\right]\medskip \\ \displaystyle \tfrac {(b-x)^2}{2}, &  \mbox{if} &  x\in \left[\displaystyle \tfrac {a+b}{2},b\right]. \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">3.3</span>
</p>
</div>
<p>Applying Theorem 2.1 for n=1, we get a result established by Ujevic [3]. </p>
<p><div class="theorem_thmwrapper " id="t4">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(f:[a,b]\to \mathbb {R}\), \(f\in C^2(a,b)\), then we have the inequality </p>
<div class="displaymath" id="25">
  \begin{align}  \label{25} & \displaystyle \tfrac {2\gamma -3S_1}{24}(b-a)^3\\ \nonumber & \le (b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)-\int _a^b f(x){\rm d}x\\ \nonumber & \le \displaystyle \tfrac {2\Gamma -3S_1}{24}(b-a)^3 \end{align}
</div>
<p> where \(\gamma ,\  \Gamma \in \mathbb {R},\) \(\gamma \le f^{(2)}(x)\le \Gamma \), \(\forall \  x\in [a,b]\) and </p>
<div class="displaymath" id="a0000000016">
  \[ S_1=\displaystyle \tfrac {f’(b)-f’(a)}{b-a}. \]
</div>
<p>If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x),\) \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities <span class="rm">(25)</span> are sharp. </p>

  </div>
</div> Similar with Theorem 2.2 and 2.3, the Gauss’s formula with a single node are given in the next theorems : </p>
<p><div class="theorem_thmwrapper " id="t5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">3.1</span> we have </p>
<div class="displaymath" id="26">
  \begin{align}  \label{26} & \displaystyle \tfrac {4\gamma -3S_1}{24}(b-a)^3\le \\ \nonumber & \le \int _a^b f(x){\rm d}x-(b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)\\ \nonumber & \le \displaystyle \tfrac {4\Gamma -3S_1}{24}(b-a)^3. \end{align}
</div>
<p> If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x)\) and \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities <span class="rm">(26)</span> are sharp. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="t6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">3.1</span> we have </p>
<div class="displaymath" id="27">
  \begin{align}  \label{27} & \displaystyle \tfrac {(b-a)^3}{24}(\gamma -2\Gamma )\le \\ \nonumber & \le (b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)-\int _a^b f(x){\rm d}x\le \\ \nonumber & \le \displaystyle \tfrac {(b-a)^3}{24}(\Gamma -2\gamma ). \end{align}
</div>
<p> If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x)\) and \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities <span class="rm">(27)</span> are sharp. </p>

  </div>
</div> Similar results, we obtain in the case of Gauss formula with two nodes. The results was established by the author in the paper [1]. </p>
<p>Let \(f:[a,b]\to \mathbb {R}\), \(f\in C^4[a,b]\) and \(x_1,x_2\in [a,b]\) so that \(x_1=\displaystyle \tfrac {a+b}{2}-\displaystyle \tfrac {b-a}{2}\cdot \xi \), \(x_2=\displaystyle \tfrac {a+b}{2}+\displaystyle \tfrac {b-a}{2}\cdot \xi \), where \(\xi =\displaystyle \tfrac {1}{\sqrt3}=0,57735027\dots .\) </p>
<p>Gauss’s quadrature formula with two nodes has the following form </p>
<div class="equation" id="28">
<p>
  <div class="equation_content">
    \begin{equation}  \label{28} \int _a^b f(x){\rm d}x=\displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]+R[f]. \end{equation}
  </div>
  <span class="equation_label">3.7</span>
</p>
</div>
<p>The error \(R[f]\) from the formula (<a href="#28">3.7</a>) is given by </p>
<div class="equation" id="29">
<p>
  <div class="equation_content">
    \begin{equation}  \label{29} R[f]=\int _a^b \varphi (x)f^{(4)}(x){\rm d}x, \end{equation}
  </div>
  <span class="equation_label">3.8</span>
</p>
</div>
<p> where the function \(\varphi \) has the form </p>
<div class="equation" id="30">
<p>
  <div class="equation_content">
    \begin{equation}  \label{30} \varphi (x)= \left\{ \begin{array}{lll} \displaystyle \tfrac {(x-a)^4}{4!}, &  \mbox{if} &  x\in [a,x_1]\\ \displaystyle \tfrac {(x-a)^4}{4!}-\displaystyle \tfrac {b-a}{2}\displaystyle \tfrac {(x-x_1)^3}{3!}, &  \mbox{if} &  x\in ]x_1,x_2[\\ \displaystyle \tfrac {(b-x)^4}{4!}, &  \mbox{if} &  x\in [x_2,b] \end{array}\right.\end{equation}
  </div>
  <span class="equation_label">3.9</span>
</p>
</div>
<p>In this case we obtain the following theorems: </p>
<p><div class="theorem_thmwrapper " id="t7">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If \(f\in C^4[a,b]\) then </p>
<div class="displaymath" id="31">
  \begin{align}  \label{31} &  \displaystyle \tfrac {1}{17280}(b-a)^5(41\gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \gamma )\\ &  \le \displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]-\int _a^b f(x){\rm d}x\nonumber \\ &  \le \displaystyle \tfrac {1}{17280}(b-a)^5 (41\Gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \Gamma )\nonumber \end{align}
</div>
<p> where \(\gamma ,\Gamma \in \mathbb {R}\), \(\gamma \le f^{(4)}(x)\le \Gamma \), for all \(x\in [a,b]\) and \(S_3=\displaystyle \tfrac {f”’(b)-f”’(a)}{b-a}\). Moreover, </p>
<div class="equation" id="32">
<p>
  <div class="equation_content">
    \begin{equation}  \label{32} \gamma =\min _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(4)}(x) \end{equation}
  </div>
  <span class="equation_label">3.11</span>
</p>
</div>
<p> the inequalities <span class="rm">(31)</span> are sharp. </p>

  </div>
</div> <div class="theorem_thmwrapper " id="t8">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">3.4</span> we have </p>
<div class="displaymath" id="33">
  \begin{align}  \label{33} & \displaystyle \tfrac {1}{17280}(b-a)^5(49\gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \gamma )\le \\ \nonumber & \le \int _a^b f(x){\rm d}x-\displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]\le \\ \nonumber & \le \displaystyle \tfrac {1}{17280}(b-a)^5(49\Gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \Gamma )\\ \nonumber \end{align}
</div>
<p> If </p>
<div class="displaymath" id="a0000000017">
  \[ \gamma =\min \limits _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max \limits _{x\in [a,b]}f^{(4)}(x) \]
</div>
<p> the inequalities <span class="rm">(33)</span> are sharp. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="t9">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the assumptions of Theorem <span class="rm">3.4</span> we have </p>
<div class="displaymath" id="34">
  \begin{align}  \label{34} & \displaystyle \tfrac {1}{34560}(b-a)^5(41\gamma -49\Gamma -180\xi ^3\gamma +180\xi ^3\Gamma )\le \\ \nonumber & \le \displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]-\int _a^b f(x){\rm d}x\le \\ \nonumber & \le \displaystyle \tfrac {1}{34560}(b-a)^5(41\Gamma -49\gamma +180\xi ^3\gamma -180\xi ^3\Gamma ) \nonumber \end{align}
</div>
<p> If </p>
<div class="displaymath" id="a0000000018">
  \[ \gamma =\min \limits _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max \limits _{x\in [a,b]}f^{(4)}(x) \]
</div>
<p> the inequalities <span class="rm">(34)</span> are sharp. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no1-art6"> <i class="sc">M. Heljiu</i>, <i class="it">Double Inequalities for quadrature formula of Gauss type with two nodes</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">XXXVII</b>, no. 1, pp.&#160;53–57, 2008. <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="2">2</a></dt>
  <dd><p><i class="sc">D. V. Ionescu</i>, <i class="it">Numerical Integration</i>, Ed. Tehnică, Bucureşti, 1957 (in Romanian). </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><i class="sc">N. Ujevic</i>, <i class="it">Some Double Integral Inequalities and Applications</i>, Acta Math. Univ. Comenianae, <b class="bf">71(2)</b>, pp.&#160;187, 2002. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><i class="sc">N. Ujevic</i>, <i class="it">Double Integral Inequalities of Simpson Type and Applications</i>, J. Appl. Math. &amp; Computing, <b class="bf">14</b>, Nos. 1–2, pp.&#160; 213–223, 2004. </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>