AN OSTROWSKI TYPE INEQUALITY FOR DOUBLE INTEGRALS IN TERMS OF L p -NORMS AND APPLICATIONS IN NUMERICAL INTEGRATION

. An inequality of the Ostrowski type for double integrals and applications in Numerical Analysis in connection with cubature formulae are given. MSC 2000. Primary: 26D15; Secondary: 41A55.


INTRODUCTION
In 1938, A. Ostrowski proved the following integral inequality [5, p. 468].The constant 1  4 is the best possible.For some generalizations see the book [5, pp. 468-484] by Mitrinović, Pecarić and Fink.Some applications of the above results in Numerical Integration and for special means have been given in [3] by S. S. Dragomir and S. Wang.
In [4] Dragomir and Wang established the following Ostrowski type inequality for differentiable mappings whose derivatives belong to L p -spaces.
Theorem 2. Let f : I ⊆ R→R be a differentiable mapping on where Note that the above inequality can also be obtained from Theorem 1 [5, p. 471] due to A. M. Fink.
be integrable and p (x) > 0, for every x ∈ D. Then for every x ∈ D, we have the inequality: In the present paper we point out an Ostrowski type inequality for double integrals in terms of L p -norms and apply it in Numerical Integration obtaining a general cubature formula.

THE RESULTS
The following inequality of Ostrowski's type for mappings of two variables holds: then we have the inequality: Proof.Integrating by parts successively, we have the equality: By similar computations, we have If we add the equalities (2) − (5) we get, in the right hand side: For the first part, let us define the kernels: Now, we deduce that the left part can be represented as: p (x, s) q (y, t) f s,t (s, t) dsdt = (6) Now, using the identity (6), we get Using Hölder's integral inequality for double integrals, we get and the theorem is proved.
Corollary 5.Under the above assumptions, we have the inequality: then, the above inequality (7) is the best that can be obtained from (1).
which is clearly equivalent to Ostrowski's inequality.Consequently (1) can be also regarded as a generalization for double integrals of the result embodied in Theorem 2.

APPLICATIONS FOR CUBATURE FORMULAE
Let us consider the arbitrary divisions for which we assume that the involved integrals can more easily be computed than the original double integral and With this assumption, we can state the following cubature formula: be as in Theorem 4 and I n , J m , ξ and η be as above.Then we have the cubature formula: where the remainder term R (f, I n , J m , ξ, η) satisfies the estimation: for all ξ and η as above.
Summing over i from 0 to n − 1 and over j from 0 to m − 1 and using the generalized triangle inequality and Hölder's discrete inequality for double sums, we deduce To prove the second part, we observe that for all i, j as above and the intermediate points ξ i and η j .We omit the details.

p
(x, s) q (y, t) f s,t (s, t) dsdt.Consequently, we get the identity

1 (h i l j f x i +x i+1 2 , y j +y j+1 2 .f
q+1) 2/q f s,t p (b − a) (d − c) ν(h)µ(l) 1 q .Now, define the sumC M (f, I n , J m ) :=Then we have the best cubature formula we can get from Theorem 6.Corollary 7.Under the above assumptions we have (s, t)dsdt = C M (f, I n , J m ) + R (f, I n , J m ) ,where the remainder R (f, I n , J m ) satisfies the estimation: R (f, I n , J m )