Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 5, 165-169 Available online at http://pubs.sciepub.com/tjant/2/5/2 © Science and Education Publishing DOI:10.12691/tjant-2-5-2 On the Simpson’s Inequality for Convex Functions on the Co-Ordinates M. EMIN ÖZDEMIR 1 , AHMET OCAK AKDEMIR 2,* , HAVVA KAVURMACI 3 1 Ataturk University, K.K. Education Faculty, Department of Mathematics, Erzurum, Turkey 2 Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey 3 Yüzüncü Yil University, Education Faculty, Department of Mathematics, Van, Turkey *Corresponding author: aocakakdemir@gmail.com Received July 28, 2014; Revised September 19, 2014; Accepted September 27, 2014 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. Keywords: Simpson’s inequality, co-ordinates, convex functions, bounded functions Cite This Article: M. EMIN ÖZDEMIR, 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, vol. 2, no. 5 (2014): 165-169. doi: 10.12691/tjant-2-5-2. 1. Introduction The following inequality is well-known in the literature as Simpson.s inequality: Theorem 1. Let [ ] : , f ab →  be a four times continuously differentiable mapping on [ ] , ab and () [ ] () ( ) 4 4 , sup . x ab f f x ∞ ∈ = <∞ Then the following inequality holds: ( ) () ( ) () ( ) 4 4 1 1 2 3 2 2 1 . 2880 b a f a f b a b f f x dx b a f b a ∞ +   +   + −     −     ≤ − ∫ For recent results on Simpson.s type inequalities see the papers [11-19]. Convexity on the co-ordinates can be given as following (see [10]); Let us consider the bidimensional interval [ ] [ ] , , ab cd ∆= × in 2  with a b < and . c d < A function : f ∆→  will be called convex on the co- ordinates if the partial mappings [ ] : , y f ab →  () ( ) , y f u f uy = and [ ] ( ) : , , x f cd u →  () ( ) , x f v f xv = are convex where defined for all [ ] , y cd ∈ and [ ] , . x ab ∈ Recall that the mapping : f ∆→  is convex on ∆ , if the following inequality; ( ) ( ) ( ) ( ) ( ) ( ) 1 , 1 , 1 , f x z y w f xy f zw λ λ λ λ λ λ + − + − ≤ + − holds for all ( )( ) , , , xy zw ∈∆ and [ ] 0,1 . λ ∈ In [10], Dragomir proved the following inequalities: Theorem 2. Suppose that [ ] [ ] : , , f ab cd ∆= × →  is convex on the co-ordinates on . ∆ Then one has the inequalities; ( )( ) ( ) ( ) ( ) ( ) ( ) 1 , , 2 2 , , , , . 4 b d a c a bc d f f x y dydx b a d c f ac f bc f ad f bd + +   ≤   − −   + + + ≤ ∫∫ (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] 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 2 : f ∆⊂ →   be a partial differentiable mapping on [ ] [ ] , , . ab cd ∆= × If ( ) 2 , f L ts ∂ ∈ ∆ ∂∂ then the following equality holds: , , 4 , 2 2 2 2 , , 2 2 9 c d c d a bc d f a f b f a b a b f c f d  + + + +        + +                   + +     + +            