A New General Inequality for double integrals

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]→R is a differentiable function with bounded derivative, then

|f(x)-(1/(b-a))∫_{a}^{b}f(t)dt|≤[(1/4)+(((x-((a+b)/2))²)/((b-a)²))](b-a)‖f′‖_{∞} <label>1</label>

for every x∈[a,b]. Moreover the constant 1/4 is the best possible. Inequality (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 (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:

K(x,y;t,s)={<K1.1/>┊

(t-(a+λ((b-a)/6)))(s-(c+λ((d-c)/6))) for a≤t≤x,c≤s≤y, (t-(a+λ((b-a)/6)))(s-(d-λ((d-c)/6))) for a≤t≤x,y≤s≤d, (t-(b-λ((b-a)/6)))(s-(c+λ((d-c)/6))) for x≤t≤b,c≤s≤y, (t-(b-λ((b-a)/6)))(s-(d-λ((d-c)/6))) for x≤t≤b,y≤s≤d. This inequality is a new generalization of the inequalities of Simpson and Ostrowski type obtained by Zhongxue in [9].