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\begin{center}
\textbf{A New General Inequality for double integrals}

\hrule

Erhan Set$^{1}$, Mehmet Zeki Sar\i kaya$^{1}$ and Ahmet Ocak Akdemir$^{2}$

$^{1}$Department of Mathematics, Faculty of Science and Arts, D\"{u}zce
University, D\"{u}zce, Turkey $\ \ $

$^{2}$A\u{g}r\i\ \.{I}brahim \c{C}e\c{c}en University, Faculty of Science
and Letters, Department of Mathematics, 04100, A\u{g}r\i , Turkey
\end{center}

\textbf{Abstract}

The inequality of Ostrowski gives us an estimate for the deviation of the
values of a smooth function from its mean value. More precisely, if $%
f:[a,b]\rightarrow \mathbb{R}$ is a differentiable function with bounded
derivative, then%
\begin{equation}
\left\vert f(x)-\frac{1}{b-a}\int\limits_{a}^{b}f(t)dt\right\vert \leq \left[
\frac{1}{4}+\frac{(x-\frac{a+b}{2})^{2}}{(b-a)^{2}}\right] (b-a)\left\Vert
f^{\prime }\right\Vert _{\infty }  \label{1}
\end{equation}%
for every $x\in \lbrack a,b]$. Moreover the constant $1/4$ is the best
possible. Inequality (\ref{1}) has wide applications in numerical analysis
and in the theory of some special means; estimating error bounds for some
special means, some mid-point, trapezoid and Simpson rules and quadrature
rules, etc. Hence inequality (\ref{1}) has attracted considerable attention
and interest from mathematicans and researchers. Due to this, over the
years, the interested reader is also refered to ([1]-[9]) for integral
inequalities in several independent variables. In addition, the current
approach of obtaining the bounds, for a particular quadrature rule, have
depended on the use of peano kernel. The general approach in the past has
involved the assumption of bounded derivatives of degree greater than one.

In this paper, we obtain a new general inequality involving functions of two
independent variables by defining the peano kernel $K(x,y;s,t)$ as the
following$:$%
\begin{equation*}
K(x,y;t,s)=\left\{ 
\begin{array}{c}
\left( t-(a+\lambda \dfrac{b-a}{6})\right) \left( s-\left( c+\lambda \dfrac{%
d-c}{6}\right) \right) \text{ \ \ \ \ for \ }a\leq t\leq x,\;c\leq s\leq y,
\\ 
\left( t-(a+\lambda \dfrac{b-a}{6})\right) \left( s-\left( d-\lambda \dfrac{%
d-c}{6}\right) \right) \text{ \ \ \ \ for \ }a\leq t\leq x,\;y\leq s\leq d,
\\ 
\left( t-(b-\lambda \dfrac{b-a}{6})\right) \left( s-\left( c+\lambda \dfrac{%
d-c}{6}\right) \right) \text{ \ \ \ \ for \ }x\leq t\leq b,\;c\leq s\leq y,
\\ 
\left( t-(b-\lambda \dfrac{b-a}{6})\right) \left( s-\left( d-\lambda \dfrac{%
d-c}{6}\right) \right) \text{ \ \ \ \ for \ }x\leq t\leq b,\;y\leq s\leq d.%
\end{array}%
\right. 
\end{equation*}%
This inequality is a new generalization of the inequalities of Simpson and
Ostrowski type obtained by Zhongxue in [9].

\textbf{References}

[1] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for
double integrals and applications for cubature formulae, Soochow J. Math.,
27(1), (2001), 109-114.

[2] M.E. \"{O}zdemir, H. Kavurmac\i\ and E. Set, Ostrowski's type
inequalities for $(\alpha ,m)-$convex functions, Kyungpook Math. J., 50
(2010), 371-378.

[3] B. G. Pachpatte, On a new Ostrowski type inequality in two independent
variables, Tamkang J. Math., 32(1), (2001), 45-49.

[4] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math.
Univ. Comenianae, Vol. LXXIX, 1(2010), pp. 129-134.

[5] M.Z. Sar\i kaya, E. Set and M.E. \"{O}zdemir, On new inequalities of
Simpson's type for s-convex functions, Computers \& Mathematics with
Applications, 60 (2010), 2191-2199.

[6] E. Set, New inequalities of Ostrowski type for mappings whose
derivatives are s-convex in the second sense via fractional integrals,
Computers and Mathematics with Applications, 63(7), (2012), 1147-1154.

[7] E. Set and M.Z. Sar\i kaya, On The Generalization Of Ostrowski And Gr%
\"{u}ss Type Discrete Inequalities, Computers \& Mathematics with
Applications, 62, (2011), 455--461.

[8] N. Ujevi\'{c}, Sharp inequalities of Simpson type and Ostrowski type,
Computers \& Mathematics with Applications, 48 (2004) 145-151.

[9] L. Zhongxue, On sharp inequalities of Simpson type and Ostrowski type in
two independent variables, Computers \& Mathematics with Applications, 56
(2008) 2043-2047.\hrule

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