Generalizations of the Simpson Like Type Inequalities for Co Ordinated

Volume 2012, Article ID 715751,16pages doi:10.1155/2012/715751

Research Article

Generalizations of the Simpson-Like Type

Inequalities for Co-Ordinated

s

-Convex Mappings

in the Second Sense

Jaekeun Park

Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea

Correspondence should be addressed to Jaekeun Park,jkpark@hanseo.ac.kr

Received 14 October 2011; Revised 6 December 2011; Accepted 21 December 2011

Academic Editor: Feng Qi

Copyrightq2012 Jaekeun Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. 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. Letf :I⊂ 0,∞ → Rbe a four-time continuous differentiable mapping ona, b

andf4∞sup_x∈a,b|f4x|<∞. Then, the following inequality holds:

1 6

fa 4f

ab

2

fb

− 1

b−a

_b

a

fxdx

≤ b−a4

2880

f4

∞.

1.1

It is well known that the mappingfis neither four times differentiable nor is the fourth

derivative f4 bounded ona, b, then we cannot apply the classical Simpson quadrature

For recent results on Simpson type inequalities, you may see the papers1–5.

In2,6–8, Dragomir et al. and Park considered among others the class of mappings

which ares-convex on the coordinates.

In the sequel, in this paper letΔ a, b×c, dbe a bidimensional interval inR2_with

a < bandc < d.

Definition 1.2. A mappingf :Δ → Rwill be calleds-convex in the second sense onΔif the following inequality:

f λx 1−λz, λy 1−λw≤λsf x, y 1−λsfz, w, 1.2

holds, for allx, y,z, w∈Δ,λ∈0,1ands∈0,1.

Modification for convex and s-convex mapping on Δ, which are also known as

co-ordinated convex, s-convex mapping, ands-r-convex, respectively, were introduced by

Dragomir, Sarikaya5,9,10, and Park4,8,11,12.

Definition 1.3. A mappingf :Δ → Rwill be called coordinateds-convex in the second sense

onΔif the partial mappings

fy :a, b−→R, fyu f u, y

, fx:c, d−→R, fxv fx, v,

1.3

ares-convex in the second sense, for allx∈a, b,y∈c, d, ands∈0,1 5,9,10.

A formal definition for coordinateds-convex mappings may be stated as follow8.

Definition 1.4. A mappingf :Δ → Rwill be called coordinateds-convex in the second sense

onΔif the following inequality:

f tx 1−tz, λy 1−λw

≤tsλsf x, y 1−tsλsf z, yts1−λsfx, w 1−ts1−λsfz, w, 1.4

holds, for allt, λ∈0,1,x, y,z, w∈Δ, ands∈0,1.

In2, S.S. Dragomir established the following theorem.

Theorem 1.5. Letf :Δ → Rbe convex on the coordinates onΔ. Then, one has the inequalities:

f

ab

2 ,

cd

2

≤ 1

b−ad−c

_b

a

_d

c

f x, ydydx

≤ 1

4

fa, c fb, c fa, d fb, d.

1.5

In13, Hwang et al. gave a refinement of Hadamard’s inequality on the coordinates

In 1, 6, 14, Alomari and Darus proved inequalities for coordinated s-convex mappings.

In15, Latif and Alomari defined coordinatedh-convex mappings, established some

inequalities for co-ordinatedh-convex mappings and proved inequalities involving product

of convex mappings on the coordinates.

In3, ¨Ozdemir et al. gave the following theorems:

Theorem 1.6. Letf :Δ⊂R2 _{→ Rbe a partial differentiable mapping onΔ. If∂}2_{f/∂t∂λis convex}

on the coordinates onΔ, then the following inequality holds:

1₉f

a,cd

2

f

b,cd

2

4f

ab

2 ,

cd

2

f

ab

2 , c

f

ab

2 , d

1

36

fa, c fa, d fb, c fb, d

1

b−ad−c

_b

a

_d

c

f x, ydydx−A

≤

5 72

2

b−ad−c

×∂2f

∂t∂λa, c

∂2_f

∂t∂λa, d

∂2_f

∂t∂λb, c

∂2_f

∂t∂λb, d

, 1.6 where

A 1 b−a

_b

a

fx, c 4fx,cd/2 fx, d

6

dx

1

d−c

_d

c

f a, y4f ab/2, yf b, y

6

dy.

1.7

Theorem 1.7. Letf:Δ⊂R2 → Rbe a partial differentiable mapping onΔ. If∂2f/∂t∂λis bounded, that is,

∂2_f

∂t∂λta 1−tb, λc 1−λd

∞ sup x,y∈a,b×c,d ∂ 2_f

∂t∂λta 1−tb, λc 1−λd

<∞

1.8

for allt, λ∈0,12, then the following inequality holds:

1₉f

a,cd

2

f

b,cd

2

4f

ab

2 ,

cd

2

f

ab

2 , c

f

ab

2 , d

1

36

1

b−ad−c a c

f x, ydydx−A

≤

5 72

2

b−ad−c∂

2_f

∂t∂λta 1−tb, λc 1−λd

∞

,

1.9

whereAis defined inTheorem 1.6.

In3, ¨Ozdemir 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 coordinateds-convex mappings in the second sense.

2. Main Results

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

Lemma 2.1. Letf : Δ → Rbe a partial differentiable mapping on Δ a, b×c, d ⊂ R2. If

∂2_{f/∂t∂λ ∈ L1Δ, then, forr}

1, r2 ≥ 2 andh1, h2 ∈ 0,1with1/r1 ≤ h1 ≤ r1−1/r1and

1/r2≤h2≤r2−1/r2, the following equality holds:

I fh1, h2, r1, r2let

r1−2r2−2

r1r2

fh1a 1−h1b, h2c 1−h2d

r1−2

r1r2

fh1a 1−h1b, c fh1a 1−h1b, d

r2−2

r1r2

fa, h2c 1−h2d fb, h2c 1−h2d

1

r1r2

fa, c fa, d fb, c fb, d

− 1

r2b−a

_b

a

fx, c r2−2fx, h2c 1−h2d fx, d

dx

− 1

r1d−c

d

c

f a, y r1−2f h1a 1−h1b, y

f b, ydy

1

b−ad−c

_b

a

_d

c

f x, ydydx

b−ad−c

₁

0

ph1, r1, tqh2, r2, λ

× ∂2f

∂t∂λta 1−tb, λc 1−λddtdλ,

where

ph1, r1, t ⎧ ⎪ ⎨ ⎪ ⎩

t− 1

r1, t∈0, h1,

t−r1−1

r1 , t∈h1,1,

qh2, r2, λ ⎧ ⎪ ⎨ ⎪ ⎩

λ− 1

r2, λ∈0, h2,

λ−r2−1

r2 , λ∈h2,1.

2.2

Proof. By the definitions ofph1, r1, tandqh2, r2, λ, we can write

I

₁

0

ph1, r1, tqh2, r2, λ∂

2_f

∂t∂λta 1−tb, λc 1−λddtdλ

1

0

qh2, r2, λ

1

0

ph1, r1, t∂

2_f

∂t∂λta 1−tb, λc 1−λddt

dλ

₁

0

qh2, r2, λ

h1

0

t− 1 r1

∂2_f

∂t∂λta 1−tb, λc 1−λddt

1

h1

t−r1−1 r1

∂2_f

∂t∂λta 1−tb, λc 1−λddt

dλ

₁

0

qh2, r2, λI11I12dλ,

2.3

where

I11

_h1

0

t− 1 r1

∂2_f

∂t∂λta 1−tb, λc 1−λddt,

I12

₁

h1

t−r1−1 r1

∂2f

∂t∂λta 1−tb, λc 1−λddt.

2.4

By integration by parts, we have

I11

h1

0

t− 1 r1

∂2_f

∂t∂λta 1−tb, λc 1−λddt

1

a−b

h1− 1

r1

∂f

∂λh1a 1−h1b, λc 1−λd

1

r1

∂f

∂λb, λc 1−λd

−

_h1

0

∂f

∂λta 1−tb, λc 1−λddt

,

I12 h1

t−r1−1 r1

∂ f

∂t∂λta 1−tb, λc 1−λddt

1

a−b

1

r1

∂f

∂λa, λc 1−λd

−

h1−r1− 1

r1

∂f

∂λh1a 1−h1b, λc 1−λd

−

1

h1

∂f

∂λta 1−tb, λc 1−λddt

.

2.6

By using the equalities2.5and2.6in2.3, we have

I

1

a−b

r1−2

r1 1

0

qh2, r2, λ∂f

∂λh1a 1−h1b, λc 1−λddλ

1

r1 1

0

qh2, r2, λ

∂f

∂λb, λc 1−λddλ

1

r1 1

0

qh2, r2, λ

∂f

∂λa, λc 1−λddλ

−

₁

0

qh2, r2, λ

₁

0

∂f

∂λta 1−tb, λc 1−λddλdt

1

a−b

r1−2

r1

I21

1

r1

I22

1

r1

I23−I24

,

2.7

where

I21

₁

0

qh2, r2, λ∂f

∂λh1a 1−h1b, λc 1−λddλ,

I22

1

0

qh2, r2, λ∂f

∂λb, λc 1−λddλ,

I23

1

0

qh2, r2, λ∂f

∂λa, λc 1−λddλ,

I24

1

0

qh2, r2, λ

1

0

∂f

∂λta 1−tb, λc 1−λddλdt.

2.8

Note that

i

h2

0

λ− 1 r2

∂f

∂λh1a 1−h1b, λc 1−λddλ

1

c−d

h2− 1

r2

fh1a 1−h1b, h2c 1−h2d

1

r2fh1

a 1−h1b, d

−

h2

0

fh1a 1−h1b, λc 1−λddλ

,

ii

1

h2

λ− r2−1 r2

∂f

∂λh1a 1−h1b, λc 1−λddλ

1

c−d

1

r2

fh1a 1−h1b, c

−

h2−r2− 1

r2

fh1a 1−h1b, h2c 1−h2d

−

1

h2

fh1a 1−h1b, λc 1−λddλ

.

2.10

By the equalities2.9and2.10, we have

I21

₁

0

qh2, r2, λ

∂f

∂λh1a 1−h1b, λc 1−λddλ

h2

0

λ− 1 r2

∂f

∂λh1a 1−h1b, λc 1−λddλ

1

h2

λ−r2−1 r2

∂f

∂λh1a 1−h1b, λc 1−λddλ

1

c−d

1

r2fh1a

1−h1b, c 1

r2fh1a

1−h1b, d

r2−2

r2 fh1a

1−h1b, h2c 1−h2d

−

₁

0

fh1a 1−h1b, λc 1−λddλ

.

2.11

By the similar way, we get the following:

I22

1

0

qh2, r2, λ∂f

∂λb, λc 1−λddλ

1

c−d

1

r2fb, c 1

r2fb, d

r2−2

r2

fb, h2c 1−h2d

−

1

0

fb, λc 1−λddλ

,

2.12

I23

1

0

qh2, r2, λ∂f

∂λa, λc 1−λddλ

1

c−d

1

r2fa, c 1

r2fa, d

r2−2

r2

fa, h2c 1−h2d

−

1

0

fa, λc 1−λddλ

,

I24 0

qh2, r2, λ 0

∂f

∂λta 1−tb, λc 1−λddλdt

1

c−d

1

r2

₁

0

fta 1−tb, cdt 1 r2

₁

0

fta 1−tb, ddt

r2−2

r2 1

0

fta 1−tb, h2c 1−h2ddt

−

1

0

fta 1−tb, λc 1−λddλdt

.

2.14

By the equalities2.7and2.11–2.14and using the change of the variablesxta

1−tbandyλc1−λdfort, λ∈0,12, then multiplying both sides withb−ad−c,

we have the required result2.1, which completes the proof.

Remark 2.2. Lemma 2.1 is a generalization of the results which proved by Sarikaya, Set, ¨

Ozdemir, and Dragomir3,5,9,10.

Theorem 2.3. Letf : Δ → R2 _{be a partial differentiable mapping on Δ a, b×c, d ⊂ R}2_.

If∂2_{f/∂t∂λis inL1Δand is a coordinateds-convex mapping in the second sense onΔ, then, for}

r1, r2≥2 andh1, h2 ∈0,1with1/r1≤h1≤r1−1/r1, and1/r2≤h2 ≤r2−1/r2the

following inequality holds:

1

b−ad−cI f

h1, h2, r1, r2

≤μ1r, s

μ2r, s∂

2_f

∂t∂λa, c

ν2r, s

∂2_f

∂t∂λa, d

ν1r, s

μ2r, s∂

2_f

∂t∂λb, c

ν2r, s

∂2_f

∂t∂λb, d

,

2.15

where

μ1r, s Mr1, s Nh1, s,

μ2r, s Mr2, s Nh2, s,

ν1r, s Mr1, s N1−h1, s,

ν2r, s Mr2, s−N1−h2, s

2.16

for

Mr, s 22r−1

s2_rs1_s−r2

s1s2rs2 ,

Nh, s h

s1_{2h−1s2h−1}

s1s2 .

Proof. FromLemma 2.1and by the coordinateds-convexity in the second sense of∂2_f/∂t∂λ, we can write

1

b−ad−cI f

h1, h2, r1, r2

≤

1

0

_{ph1, r1, tqh2, r2, λ}

×∂2f

∂t∂λta 1−tb, λc 1−λd

dtdλ

≤

₁

0

_{ph1, r1, tqh2, r2, λ}

×

ts_λs

∂

2_f

∂t∂λa, c

ts1−λs

∂

2_f

∂t∂λa, d

1−tsλs∂

2_f

∂t∂λb, c

1−ts1−λs

∂2_f

∂t∂λb, d

dtdλ

1

0

_{ph1, r1, tt}s_dt

1

0

_{qh2, r2, λλ}s_dλ

∂

2_f

∂t∂λa, c

₁

0

_{qh2, r2, λ1−λ}s_dλ

∂2_f

∂t∂λa, d

1

0

_{ph1, r1, t1−t}s

dt

1

0

_{qh2, r2, λλ}s_dλ

∂2_f

∂t∂λb, c

₁

0

_{qh2, r2, λ1−λ}s_dλ

∂2_f

∂t∂λb, d

.

2.18

Note that

i

₁

0

_{ph1, r1, tt}s_{dtμ1r, s,}

ii

₁

0

_{ph1, r1, t1−t}s_{dtν1r, s,}

iii

₁

0

_{qh2, r2, λλ}s_{dλμ2r, s,}

iv

₁

0

_{qh2, r2, λ1−λ}s_{dλν2r, s.}

2.19

Remark 2.4. InTheorem 2.3,

iif we chooseh1h21/2,r1r26, ands1 in2.15, then we get

I f1

2,

1

2,6,6

≤

5 72

2

Mb−ad−c, 2.20

iiif we chooseh1h21/2, r1r22, ands1 in2.15, then we get

I f1

2,

1

2,2,2

≤

1 8

2

Mb−ad−c, 2.21

where

M∂

2_f

∂t∂λa, c

∂2_f

∂t∂λa, d

∂2_f

∂t∂λb, c

∂2_f

∂t∂λb, d

, 2.22

which implies thatTheorem 2.3is a generalization ofTheorem 1.6.

Theorem 2.5. Letf : Δ → R2 _{be a partial differentiable mapping onΔ a, b×c, d ⊂ R}2_{. If}

∂2_{f/∂t∂λis bounded, that is,}

∂2_f

∂t∂λ

∞

let

∂2_f

∂t∂λta 1−tb, λc 1−λd

∞

sup

x,y∈a,b×c,d

∂2_f

∂t∂λta 1−tb, λc 1−λd

<∞

2.23

for allt, λ∈0,12, then, forr1, r2 ≥2 andh1, h2 ∈0,1with1/r1≤h1 ≤ r1−1/r1and 1/r2≤h2≤r2−1/r2, the following inequality holds:

1

b−ad−cI f

h1, h2, r1, r2

≤

1

2−h1h

2

1

2−r1

r2 1

1

2 −h2h

2

2

2−r2

r2 2

∂

2_f

∂t∂λ

∞

.

2.24

Proof. FromLemma 2.1, using the property of modulus and the boundedness of∂2_{f/∂t∂s, we} get

1

b−ad−cI f

h1, h2, r1, r2

≤∂2f

∂t∂λ

∞

₁

0

By the simple calculations, we have

i

1

0

_{ph1, r1, tdt} 1

2 −h1h

2

1

2−r1

r₁2 , 2.26

ii

₁

0

_{qh2, r2, λdt} 1

2−h2h

2

2

2−r2

r2 2

. 2.27

By using the inequality 2.25 and the equalities 2.26-2.27, the assertion 2.24

holds.

Remark 2.6. InTheorem 2.5,

iif we chooseh1h2 1

2 andr1r26, then we get

I f1

2,

1

2,6,6

≤

5 36

2

∂2_f

∂t∂λ

∞

b−ad−c, 2.28

iiif we chooseh1h21/2 andr1r22, then we get

I f1

2,

1

2,2,2

≤

1 4

2

∂2_f

∂t∂λ

∞

b−ad−c, 2.29

which implies thatTheorem 2.5is a generalization ofTheorem 1.7.

The following theorem is a generalization ofTheorem 1.6.

Theorem 2.7. Letf : Δ a, b×c, d ⊂ R2 _{→ R}2 _{be a partial differentiable mapping on} Δ a, b×c, d. If|∂2_f/∂t∂λ|q_{q >1is inL1Δand is a coordinateds-convex mapping in the}

second sense onΔ, then, forr1, r2 ≥ 2 andh1, h2 ∈ 0,1with1/r1 ≤ h1 ≤ r1−1/r1and 1/r2≤h2≤r2−1/r2, the following inequality holds:

1

b−ad−cI f

h1, h2, r1, r2≤μ1/p₃ ν₃1/p

×∂2f/∂t∂λa, c

q

∂2_{f/∂t∂λa, d}q_∂2_{f/∂t∂λb, c}q_∂2_{f/∂t∂λb, d}q

s12

1/p

,

where

μ3

2 r1−r1h1−1p1 r1h1−1p1

r₁p1 p1 ,

ν3 2 r2−r2h2−1

p1 _r

2h2−1p1

r₂p1 p1 .

2.31

Proof. FromLemma 2.1, we can write

1

b−ad−cI f

h1, h2, r1, r2

≤

₁

0

_{ph1, r1, tqh2, r2, λ}

×∂2f

∂t∂λta 1−tb, λc 1−λd

dtdλ

≤

1

0

_{ph1, r1, tqh2, r2, λ}p

dtdλ

1/p

×

1

0

∂

2_f

∂t∂λta 1−tb, λc 1−λd

q

dtdλ

1/q

.

2.32

Hence, by the inequality2.32and the coordinateds-convexity in the second sense of

|∂2_f/∂t∂λ|q_{, it follows that}

1

b−ad−cI f

h1, h2, r1, r2

≤

1

0

_{ph1, r1, tqh2, r2, λ}p

dtdλ

1/p

×

|∂2_{f/∂t∂λa, c|}q_|∂2_{f/∂t∂λa, d|}q_|∂2_{f/∂t∂λb, c|}q_|∂2_{f/∂t∂λb, d|}q

s12

1/q

.

2.33

Note that

i

₁

0

_{ph1, r1, t}p

dt 2 r1−r1h1−1

p1 _r

1h1−1p1

r₁p1 p1 , 2.34

ii

₁

0

_{qh2, r2, λ}p

dλ 2 r2−r2h2−1

p1 _r

2h2−1p1

r₂p1 p1 . 2.35

Remark 2.8. InTheorem 2.7,

iif we chooseh1h21/2,r1r26, ands1, then we get

I f1

2,

1

2,6,6

≤

2 12p1

6p1 p1

2/p

b−ad−cMq1/q, 2.36

iiif we chooseh1h21/2,r1r22, ands1, then we get

I f1

2,

1

2,2,2

≤

1

2p _p1

2/p

b−ad−cM1/qq , 2.37

iiiif we chooseh1h21/2,r1r26,s1, andq1, then we get

I f1

2,

1

2,6,6

≤

5 36

2

b−ad−cM1, 2.38

where

Mq

_∂2_{f/∂t∂λa, c}q_∂2_{f/∂t∂λa, d}q_∂2_{f/∂t∂λb, c}q_∂2_{f/∂t∂λb, d}q

4 . 2.39

Theorem 2.9. Letf : Δ → R2 _{be a partial differentiable mapping onΔ a, b×c, d ⊂ R}2_{. If}

|∂2_f/∂t∂λ|q_{q≥1is inL1Δand is a coordinateds-convex mapping in the second sense onΔ, then,}

forr1, r2 ≥2 andh1, h2 ∈0,1with1/r1≤h1 ≤r1−1/r1and1/r2≤h2≤r2−1/r2,

the following inequality holds:

1

b−ad−cI f

h1, h2, r1, r2

≤

1

2 −h1h

2

1

2−r1

r₁2

1

2 −h2h

2

2

2−r2

r₂2

_1−1/q

×

μ1

μ2

∂2_f

∂t∂λa, c

q

ν2

∂2_f

∂t∂λa, d

q

ν1

μ2

∂2_f

∂t∂λb, c

q

ν2

∂2_f

∂t∂λb, d

q1/q

,

2.40

Proof. FromLemma 2.1, we can write

1

b−ad−cI f

h1, h2, r1, r2

≤

1

0

_{ph1, r1, tqh2, r2, λ}

×∂2f

∂t∂λta 1−tb, λc 1−λd

dtdλ

≤

1

0

_{ph1, r1, tqh2, r2, λdtdλ}1−1/q

×

1

0

_{ph1, r1, tqh2, r2, λ}

×∂2f

∂t∂λta 1−tb, λc 1−λd

q

dtdλ

1/q

.

2.41

By the simple calculations, we have

i

1

0

_{ph1, r1, tdt} 1

2 −h1h

2

1

2−r1

r₁2 , 2.42

ii

₁

0

_{qh2, r2, λdλ} 1

2 −h2h

2

2

2−r2

r2 2

. 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, fort∈0,1,

1

0

_{ph1, r1, tqh2, r2, λ}

∂2_f

∂t∂λta 1−tb, λc 1−λd

q

dtdλ

≤

1

0

_{ph1, r1, tqh2, r2, λ}

×

ts_λs

∂

2_f

∂t∂λa, c

q

ts_1−λs

∂

2_f

∂t∂λa, d

q

1−tsλs∂

2_f

∂t∂λb, c

q

1−ts1−λs∂

2_f

∂t∂λb, d

q

1

0

_{ph1, r1, tt}s_dt

1

0

_{qh2, r2, λλ}s_dλ

∂2_f

∂t∂λa, c

q

1

0

_{qh2, r2, λ1−λ}s_dλ

∂2_f

∂t∂λa, d

q

1

0

_{ph1, r1, t1−t}s

dt

1

0

_{qh2, r2, sλ}s_dλ

∂2_f

∂t∂sb, c

q

1

0

_{qh2, r2, λ1−λ}s_dλ

∂2_f

∂t∂λb, d

q

μ1

μ2

∂

2_f

∂t∂λa, c

q

ν2

∂

2_f

∂t∂λa, d

q

ν1

μ2

∂2_f

∂t∂λb, c

q

ν2

∂2_f

∂t∂λb, d

q

. 2.44

By2.41–2.44, the assertion2.40holds.

References

1 M. Alomari, M. Darus, and S. S. Dragomir, “New inequalities of Simpson’s type for s-convex functions with applications,” RGMIA Research Report Collection, vol. 12, no. 4, article 9, 2009.

2 S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson’s inequality and applications,” Journal of

Inequalities and Applications, vol. 5, no. 6, pp. 533–579, 2000.

3 M. E. ¨Ozdemir, A. O. Akdemir, H. Kavurmaci, and M. Avci, “On the Simpson’s inequality for co-ordinated convex functions,” Classical Analysis and ODEs,http://arxiv.org/abs/1101.0075.

4 J. Park, “Generalization of some Simpson-like type inequalities via differentiable s-convex mappings in the second sense,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 493531, 13 pages, 2011.

5 M. Z. Sarikaya, E. Set, and M. E. Ozdemir, “On new inequalities of Simpson’s type for s-convex functions,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2191–2199, 2010.

6 M. Alomari and M. Darus, “Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 29-32, pp. 1557–1567, 2008.

7 S. S. Dragomir, “On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane,” Taiwanese Journal of Mathematics, vol. 5, no. 5, pp. 775–788, 2001.

8 J. Park, “Generalizations of the Simpson-like type inequalities for coordinated s-convex mappings,”

Far East Journal of Mathematical Sciences, vol. 54, no. 2, pp. 225–236, 2011.

9 M. Z. Sarikaya, E. Set, M. E. Ozdemir, and S. S. Dragomir, “New some Hadamard’s type inequalities for co-ordinated convex functions,” Classical Analysis and ODEs,http://arxiv.org/abs/1005.0700.

10 E. Set, M. E. Ozdemir, and M. Z. Sarikaya, “Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are mconvex,” AIP Conference Proceedings, vol. 1309, no. 1, pp. 861– 873, 2010.

11 J. Park, “New generalizations of Simpson’s type inequalities for twice differentiable convex mappings,” Far East Journal of Mathematical Sciences, vol. 52, no. 1, pp. 43–55, 2011.

12 J. Park, “Some Hadamard’s type inequalities for co-ordinateds, m-convex mappings in the second sense,” Far East Journal of Mathematical Sciences, vol. 51, no. 2, pp. 205–216, 2011.

14 M. Alomari and M. Darus, “The Hadamard’s inequality for s-convex function of 2-variables on the co-ordinates,” International Journal of Mathematical Analysis, vol. 2, no. 13-16, pp. 629–638, 2008.

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