On the Simpson’s Inequality for Convex Functions on the Co-Ordinates

M. EMIN ÖZDEMIR, AHMET OCAK AKDEMIR, HAVVA KAVURMACI

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On the Simpson’s Inequality for Convex Functions on the Co-Ordinates

M. EMIN ÖZDEMIR1, AHMET OCAK AKDEMIR2,, HAVVA KAVURMACI3

1Ataturk University, K.K. Education Faculty, Department of Mathematics, Erzurum, Turkey

2Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey

3Yüzüncü Yil University, Education Faculty, Department of Mathematics, Van, Turkey

 

Abstract

In this paper, a new lemma is proved and inequalities of Simpson type are established for convex functions on the co-ordinates and bounded functions.

Cite this article:

  • ÖZDEMIR, M. EMIN, AHMET OCAK AKDEMIR, and HAVVA KAVURMACI. "On the Simpson’s Inequality for Convex Functions on the Co-Ordinates." Turkish Journal of Analysis and Number Theory 2.5 (2014): 165-169.
  • ÖZDEMIR, M. E. , AKDEMIR, A. O. , & KAVURMACI, H. (2014). On the Simpson’s Inequality for Convex Functions on the Co-Ordinates. Turkish Journal of Analysis and Number Theory, 2(5), 165-169.
  • ÖZDEMIR, M. EMIN, AHMET OCAK AKDEMIR, and HAVVA KAVURMACI. "On the Simpson’s Inequality for Convex Functions on the Co-Ordinates." Turkish Journal of Analysis and Number Theory 2, no. 5 (2014): 165-169.

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1. Introduction

The following inequality is well-known in the literature as Simpson.s inequality:

Theorem 1. Let be a four times continuously differentiable mapping on and Then the following inequality holds:

For recent results on Simpson.s type inequalities see the papers [11-19][11].

Convexity on the co-ordinates can be given as following (see [10]);

Let us consider the bidimensional interval in with and A function will be called convex on the co-ordinates if the partial mappings and are convex where defined for all and

Recall that the mapping is convex on , if the following inequality;

holds for all and

In [10], Dragomir proved the following inequalities:

Theorem 2. Suppose that is convex on the co-ordinates on Then one has the inequalities;

(1.1)

The above inequalities are sharp.

Recently, several papers have been written on the convex functions on the co-ordinates. Similar results can be found in [1-9][1] and [20, 21, 22, 23].

In this paper, we will give Simpson-type inequalities for convex functions on the co-ordinates and bounded functions on the basis of the following lemma.

2. Main Results

To prove our main result, we need the following lemma.

Lemma 1. Let be a partial differentiable mapping on If then the following equality holds:

(2.1)

where

and

Proof. Integrating by parts, we can write

By integrating the right hand side of equality, we get

Computing these integrals and using the change of the variable and for then multiplying both sides with we get the desired result.

Theorem 3. Let be a partial differentiable mapping If is a convex function on the co-ordinates on and ,then the following inequality holds:

where

Proof. By using Lemma 1, we can write

Since is co-ordinated convex on ,we get

Computing the integral in the right hand side of above inequality, we have

We obtain

(2.2)

By a similar argument for the above integral, we have

(2.3)

If we use (2.3) in (2.2), we get the required result.

Theorem 4. Let be a partial differentiable mapping on If is bounded, i.e.,

for all and Then the following inequality holds:

where A is as in Theorem 3.

Proof. From Lemma 1 and using the property of modulus, we have

Since is bounded, we have

(2.4)

By a simple calculation,

(2.5)

If we use (2.5) in (2.4), we have

This completes the proof.

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