Generalizations of the Simpson-Like Type Inequalities for Co-Ordinated s-Convex Mappings in the Second Sense

A generalized identity for some partial differentiable mappings on a bidimensional interval is obtained, and, by using this result, the author establishes generalizations of Simpson-like type inequalities for coordinated s-convex mappings in the second sense.


Introduction
In recent years, a number of authors have considered error estimate inequalities for some known and some new quadrature formulas.Sometimes they have considered generalizations of the Simpson-like type inequality which gives an error bound for the well-known Simpson rule.
Theorem 1.1.Let f : I ⊂ 0, ∞ → R be a four-time continuous differentiable mapping on a, b and f 4 ∞ sup x∈ a,b |f 4 x | < ∞.Then, the following inequality holds: It is well known that the mapping f is neither four times differentiable nor is the fourth derivative f 4 bounded on a, b , then we cannot apply the classical Simpson quadrature formula.

International Journal of Mathematics and Mathematical Sciences
For recent results on Simpson type inequalities, you may see the papers 1-5 .In 2, 6-8 , Dragomir et al. and Park considered among others the class of mappings which are s-convex on the coordinates.
In the sequel, in this paper let Δ a, b × c, d be a bidimensional interval in R 2 with a < b and c < d.Definition 1.2.A mapping f : Δ → R will be called s-convex in the second sense on Δ if the following inequality: holds, for all x, y , z, w ∈ Δ, λ ∈ 0, 1 and s ∈ 0, 1 .
Modification for convex and s-convex mapping on Δ, which are also known as co-ordinated convex, s-convex mapping, and s-r-convex, respectively, were introduced by Dragomir, Sarikaya 5, 9, 10 , and Park 4,8,11,12 .Definition 1.3.A mapping f : Δ → R will be called coordinated s-convex in the second sense on Δ if the partial mappings are s-convex in the second sense, for all x ∈ a, b , y ∈ c, d , and s ∈ 0, 1 5, 9, 10 .
A formal definition for coordinated s-convex mappings may be stated as follow 8 .
In 2 , S.S. Dragomir established the following theorem.
Theorem 1.5.Let f : Δ → R be convex on the coordinates on Δ.Then, one has the inequalities: 1.5 In 13 , Hwang et al. gave a refinement of Hadamard's inequality on the coordinates and they proved some inequalities for coordinated convex mappings.

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In 1, 6, 14 , Alomari and Darus proved inequalities for coordinated s-convex mappings.
In 15 , Latif and Alomari defined coordinated h-convex mappings, established some inequalities for co-ordinated h-convex mappings and proved inequalities involving product of convex mappings on the coordinates.
In 3 , Özdemir et al. gave the following theorems: Theorem 1.6.Let f : Δ ⊂ R 2 → R be a partial differentiable mapping on Δ.If ∂ 2 f/∂t∂λ is convex on the coordinates on Δ, then the following inequality holds: where 4f a b /2, y f b, y 6 dy.
for all t, λ ∈ 0, 1 2 , then the following inequality holds: where A is defined in Theorem 1.6.
In 3 , Özdemir et al. proved a new equality and, by using this equality, established some inequalities on coordinated convex mappings.
In this paper the author give a generalized identity for some partial differentiable mappings on a bidimensional interval and, by using this result, establish a generalizations of Simpson-like type inequalities for coordinated s-convex mappings in the second sense.

Main Results
To prove our main results, we need the following lemma.

2.2
Proof.By the definitions of p h 1 , r 1 , t and q h 2 , r 2 , λ , we can write where

2.4
By integration by parts, we have International Journal of Mathematics and Mathematical Sciences

2.6
By using the equalities 2.5 and 2.6 in 2.3 , we have where

2.8
Note that i International Journal of Mathematics and Mathematical Sciences

2.10
By the equalities 2.9 and 2.10 , we have

2.11
By the similar way, we get the following: 2.12

2.13
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2.14
By the equalities 2.7 and 2.11 -2.14 and using the change of the variables x ta 1−t b and y λc 1−λ d for t, λ ∈ 0, 1 2 , then multiplying both sides with b − a d − c , we have the required result 2.1 , which completes the proof.

2.17
Proof.From Lemma 2.1 and by the coordinated s-convexity in the second sense of ∂ 2 f/∂t∂λ, we can write

2.18
Note that i ii if we choose h 1 h 2 1/2, r 1 r 2 2, and s 1 in 2.15 , then we get where which implies that Theorem 2.3 is a generalization of Theorem 1.6.

2.24
Proof.From Lemma 2.1, using the property of modulus and the boundedness of

2.25
By the simple calculations, we have

2.27
By using the inequality 2.25 and the equalities 2.26 -2.27 , the assertion 2.24 holds.
Remark 2.6.In Theorem 2.5, i if we choose h 1 h 2 1 2 and r 1 r 2 6, then we get ii if we choose h 1 h 2 1/2 and r 1 r 2 2, then we get which implies that Theorem 2.5 is a generalization of Theorem 1.7.
The following theorem is a generalization of Theorem 1.6.

2.31
Proof.From Lemma 2.1, we can write Hence, by the inequality 2.32 and the coordinated s-convexity in the second sense of |∂ 2 f/∂t∂λ| q , it follows that ii if we choose h 1 h 2 1/2, r 1 r 2 2, and s 1, then we get 2.37 iii if we choose h 1 h 2 1/2, r 1 r 2 6, s 1, and q 1, then we get where and is a coordinated s-convex mapping in the second sense on Δ, then, for r 1 , r 2 ≥ 2 and h 1 , h 2 ∈ 0, 1 with 1/r 1 ≤ h 1 ≤ r 1 − 1 /r 1 and 1/r 2 ≤ h 2 ≤ r 2 − 1 /r 2 , the following inequality holds:

2.43
Since |∂ 2 f/∂t∂λ| q is a coordinated s-convex mapping in the second sense on Δ a, b × c, d , we have that, for t ∈ 0, 1 , International Journal of Mathematics and Mathematical SciencesRemark 2.4.In Theorem 2.3, i if we choose h 1 h 2 1/2, r 1 r 2 6, and s 1 in 2.15 , then we get s .2.19By 2.18 and 2.19 , we get the inequality 2.15 by the simple calculations.