Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 715751, 16 pages
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
Copyright q 2012 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. Let f : I ⊂ 0, ∞ → R be a four-time continuous differentiable mapping on a, b
and f 4 ∞ supx∈a,b |f 4 x| < ∞. Then, the following inequality holds:
b
1
ab
1
fxdx
fb −
fa 4f
6
2
b−a a
b − a4 4 ≤
f .
∞
2880
1.1
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.
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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 R2 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:
f λx 1 − λz, λy 1 − λw ≤ λs f x, y 1 − λs fz, w,
1.2
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
fy : a, b −→ R,
fx : c, d −→ R,
fy u f u, y ,
fx v fx, v,
1.3
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.
Definition 1.4. A mapping f : Δ → R will be called coordinated s-convex in the second sense
on Δ if the following inequality:
f tx 1 − tz, λy 1 − λw
≤ ts λs f x, y 1 − ts λs f z, y ts 1 − λs fx, w 1 − ts 1 − λs fz, w,
1.4
holds, for all t, λ ∈ 0, 1, x, y, z, w ∈ Δ, and s ∈ 0, 1.
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:
b d
1
ab cd
,
≤
f
f x, y dydx
2
2
b − ad − c a c
1
≤ fa, c fb, c fa, d fb, d .
4
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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3
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 ⊂ R2 → 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:
1
f a, c d f b, c d 4f a b , c d
9
2
2
2
2
ab
1
ab
,c f
,d
f
fa, c fa, d fb, c fb, d
2
2
36
b d
1
f x, y dydx − A
b − ad − c a c
2
5
≤
b − ad − c
72
∂2 f
∂2 f
∂2 f
∂2 f
× a, c a, d b, c b, d ,
∂t∂λ
∂t∂λ
∂t∂λ
∂t∂λ
1.6
where
b
fx, c 4fx, c d/2 fx, d
dx
6
a
d
f a, y 4f a b/2, y f b, y
1
dy.
d−c c
6
1
A
b−a
1.7
Theorem 1.7. Let f : Δ ⊂ R2 → R be a partial differentiable mapping on Δ. If ∂2 f/∂t∂λ is bounded,
that is,
∂2 f
ta 1 − tb, λc 1 − λd
∂t∂λ
∞
∂2 f
sup
ta 1 − tb, λc 1 − λd < ∞
x,y∈a,b×c,d ∂t∂λ
for all t, λ ∈ 0, 12 , then the following inequality holds:
1
f a, c d f b, c d 4f a b , c d f a b , c f a b , d
9
2
2
2
2
2
2
1
fa, c fa, d fb, c fb, d
36
1.8
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International Journal of Mathematics and Mathematical Sciences
b d
1
f x, y dydx − A
b − ad − c a c
2
∂2 f
5
≤
b − ad − c
ta 1 − tb, λc 1 − λd ,
∂t∂λ
72
∞
1.9
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.
2. Main Results
To prove our main results, we need the following lemma.
Lemma 2.1. Let f : Δ → R be a partial differentiable mapping on Δ a, b × c, d ⊂ R2 . If
∂2 f/∂t∂λ ∈ L1 Δ, then, for r1 , r2 ≥ 2 and h1 , h2 ∈ 0, 1 with 1/r1 ≤ h1 ≤ r1 − 1/r1 and
1/r2 ≤ h2 ≤ r2 − 1/r2 , the following equality holds:
r1 − 2r2 − 2
I f h1 , h2 , r1 , r2 let
fh1 a 1 − h1 b, h2 c 1 − h2 d
r1 r2
r1 − 2 fh1 a 1 − h1 b, c fh1 a 1 − h1 b, d
r1 r2
r2 − 2 fa, h2 c 1 − h2 d fb, h2 c 1 − h2 d
r1 r2
1 fa, c fa, d fb, c fb, d
r1 r2
b
1
−
fx, c r2 − 2fx, h2 c 1 − h2 d fx, d dx
r2 b − a a
d
1
−
f a, y r1 − 2f h1 a 1 − h1 b, y f b, y dy
r1 d − c c
b d
1
f x, y dydx
b − ad − c a c
1
b − ad − c
ph1 , r1 , tqh2 , r2 , λ
0
∂2 f
×
ta 1 − tb, λc 1 − λddtdλ,
∂t∂λ
2.1
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5
where
⎧
1
⎪
⎨t − ,
r
1
ph1 , r1 , t ⎪
⎩t − r1 − 1 ,
r1
⎧
1
⎪
⎨λ − ,
r2
qh2 , r2 , λ ⎪
⎩λ − r2 − 1 ,
r2
t ∈ 0, h1 ,
t ∈ h1 , 1,
λ ∈ 0, h2 ,
λ ∈ h2 , 1.
2.2
Proof. By the definitions of ph1 , r1 , t and qh2 , r2 , λ, we can write
1
∂2 f
ph1 , r1 , tqh2 , r2 , λ
ta 1 − tb, λc 1 − λddtdλ
∂t∂λ
0
1
1
∂2 f
qh2 , r2 , λ
ph1 , r1 , t
ta 1 − tb, λc 1 − λddt dλ
∂t∂λ
0
0
h 1
1
1 ∂2 f
qh2 , r2 , λ
t−
ta 1 − tb, λc 1 − λddt
r1 ∂t∂λ
0
0
1 r1 − 1 ∂2 f
t−
ta 1 − tb, λc 1 − λddt dλ
r1
∂t∂λ
h1
1
qh2 , r2 , λI11 I12 dλ,
I
2.3
0
where
∂2 f
ta 1 − tb, λc 1 − λddt,
∂t∂λ
0
1 r1 − 1 ∂2 f
t−
ta 1 − tb, λc 1 − λddt.
r1
∂t∂λ
h1
I11 I12
h1 t−
1
r1
2.4
By integration by parts, we have
h1 I11
1 ∂2 f
t−
ta 1 − tb, λc 1 − λddt
r1 ∂t∂λ
0
1 ∂f
1
h1 −
h1 a 1 − h1 b, λc 1 − λd
a−b
r1 ∂λ
1 ∂f
b, λc 1 − λd
r1 ∂λ
h1
∂f
−
ta 1 − tb, λc 1 − λddt ,
0 ∂λ
2.5
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1 r1 − 1 ∂2 f
t−
ta 1 − tb, λc 1 − λddt
r1
∂t∂λ
h1
1 ∂f
1
a, λc 1 − λd
a−b
r1 ∂λ
r1 − 1 ∂f
− h1 −
h1 a 1 − h1 b, λc 1 − λd
r1
∂λ
1
∂f
−
ta 1 − tb, λc 1 − λddt .
h1 ∂λ
I12 2.6
By using the equalities 2.5 and 2.6 in 2.3, we have
1
∂f
r1 − 2
qh2 , r2 , λ h1 a 1 − h1 b, λc 1 − λddλ
I
r1
∂λ
0
1
∂f
1
qh2 , r2 , λ b, λc 1 − λddλ
r1
∂λ
0
1
∂f
1
qh2 , r2 , λ a, λc 1 − λddλ
r1
∂λ
0
1
1
∂f
− qh2 , r2 , λ
ta 1 − tb, λc 1 − λddλdt
0
0 ∂λ
r1 − 2
1
1
1
I21 I22 I23 − I24 ,
a−b
r1
r1
r1
1
a−b
2.7
where
I21 1
qh2 , r2 , λ
∂f
h1 a 1 − h1 b, λc 1 − λddλ,
∂λ
qh2 , r2 , λ
∂f
b, λc 1 − λddλ,
∂λ
0
I22 1
0
1
I23
I24
∂f
qh2 , r2 , λ a, λc 1 − λddλ,
∂λ
0
1
1
∂f
qh2 , r2 , λ
ta 1 − tb, λc 1 − λddλdt.
0
0 ∂λ
2.8
Note that
i
h2 1 ∂f
h1 a 1 − h1 b, λc 1 − λddλ
r2 ∂λ
0
1
1
fh1 a 1 − h1 b, h2 c 1 − h2 d
h2 −
c−d
r2
1
fh1 a 1 − h1 b, d
r2
h2
fh1 a 1 − h1 b, λc 1 − λddλ ,
−
λ−
0
2.9
International Journal of Mathematics and Mathematical Sciences
1 r2 − 1 ∂f
ii
h1 a 1 − h1 b, λc 1 − λddλ
r2
∂λ
h2
1
1
fh1 a 1 − h1 b, c
c−d
r2
r2 − 1
− h2 −
fh1 a 1 − h1 b, h2 c 1 − h2 d
r2
1
7
λ−
2.10
fh1 a 1 − h1 b, λc 1 − λddλ .
−
h2
By the equalities 2.9 and 2.10, we have
I21 1
qh2 , r2 , λ
0
∂f
h1 a 1 − h1 b, λc 1 − λddλ
∂λ
h2 1 ∂f
λ−
h1 a 1 − h1 b, λc 1 − λddλ
r2 ∂λ
0
1 r2 − 1 ∂f
λ−
h1 a 1 − h1 b, λc 1 − λddλ
r2
∂λ
h2
1
1
1
fh1 a 1 − h1 b, c fh1 a 1 − h1 b, d
c − d r2
r2
r2 − 2
fh1 a 1 − h1 b, h2 c 1 − h2 d
r2
1
2.11
fh1 a 1 − h1 b, λc 1 − λddλ .
−
0
By the similar way, we get the following:
1
∂f
qh2 , r2 , λ b, λc 1 − λddλ
∂λ
0
1
1
r2 − 2
1
fb, c fb, d fb, h2 c 1 − h2 d
c − d r2
r2
r2
1
I22 2.12
fb, λc 1 − λddλ ,
−
0
1
∂f
qh2 , r2 , λ a, λc 1 − λddλ
∂λ
0
1
r2 − 2
1
1
fa, c fa, d fa, h2 c 1 − h2 d
c − d r2
r2
r2
1
I23 fa, λc 1 − λddλ ,
−
0
2.13
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1
∂f
qh2 , r2 , λ
ta 1 − tb, λc 1 − λddλdt
0
0 ∂λ
1
1
1 1
1
fta 1 − tb, cdt fta 1 − tb, ddt
c − d r2 0
r2 0
1
r2 − 2
fta 1 − tb, h2 c 1 − h2 ddt
r2
0
1
I24 1
2.14
fta 1 − tb, λc 1 − λddλdt .
−
0
By the equalities 2.7 and 2.11–2.14 and using the change of the variables x ta 1−tb and y λc 1−λd for t, λ ∈ 0, 12 , then multiplying both sides with b − ad − c,
we have the required result 2.1, which completes the proof.
Remark 2.2. Lemma 2.1 is a generalization of the results which proved by Sarikaya, Set,
Özdemir, and Dragomir 3, 5, 9, 10.
Theorem 2.3. Let f : Δ → R2 be a partial differentiable mapping on Δ a, b × c, d ⊂ R2 .
If ∂2 f/∂t∂λ is in L1 Δ and is a coordinated s-convex mapping in the second sense on Δ, then, for
r1 , r2 ≥ 2 and h1 , h2 ∈ 0, 1 with 1/r1 ≤ h1 ≤ r1 − 1/r1 , and 1/r2 ≤ h2 ≤ r2 − 1/r2 the
following inequality holds:
1
I f h1 , h2 , r1 , r2 b − ad − c
∂2 f
∂2 f
≤ μ1 r, s μ2 r, s
a, c ν2 r, s
a, d
∂t∂λ
∂t∂λ
∂2 f
∂2 f
ν1 r, s μ2 r, s
b, c ν2 r, s
b, d ,
∂t∂λ
∂t∂λ
2.15
where
μ1 r, s Mr1 , s Nh1 , s,
μ2 r, s Mr2 , s Nh2 , s,
ν1 r, s Mr1 , s N1 − h1 , s,
2.16
ν2 r, s Mr2 , s − N1 − h2 , s
for
2 2r − 1s2 r s1 s − r 2
,
s 1s 2r s2
hs1 2h − 1s 2h − 1
Nh, s .
s 1s 2
Mr, s 2.17
International Journal of Mathematics and Mathematical Sciences
9
Proof. From Lemma 2.1 and by the coordinated s-convexity in the second sense of ∂2 f/∂t∂λ,
we can write
1
I f h1 , h2 , r1 , r2 b − ad − c
1
ph1 , r1 , tqh2 , r2 , λ
≤
0
∂2 f
×
ta 1 − tb, λc 1 − λddtdλ
∂t∂λ
1
ph1 , r1 , tqh2 , r2 , λ
≤
0
2
2
s
s ∂ f
s s ∂ f
× tλ
a, c t 1 − λ a, d
∂t∂λ
∂t∂λ
∂2 f
2
s ∂ f
s s
s
1 − t λ b, c 1 − t 1 − λ b, d dtdλ
∂t∂λ
∂t∂λ
1
1
2
ph1 , r1 , tts dt
qh2 , r2 , λλs dλ ∂ f a, c
∂t∂λ
0
0
1
∂2 f
s
qh2 , r2 , λ1 − λ dλ
a, d
∂t∂λ
0
1
1
2
ph1 , r1 , t1 − ts dt
qh2 , r2 , λλs dλ ∂ f b, c
∂t∂λ
0
0
1
∂2 f
s
qh2 , r2 , λ1 − λ dλ
b, d .
∂t∂λ
0
2.18
Note that
i
1
ph1 , r1 , tts dt μ1 r, s,
0
ii
1
ph1 , r1 , t1 − ts dt ν1 r, s,
0
iii
1
qh2 , r2 , λλs dλ μ2 r, s,
0
iv
1
qh2 , r2 , λ1 − λs dλ ν2 r, s.
0
By 2.18 and 2.19, we get the inequality 2.15 by the simple calculations.
2.19
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International Journal of Mathematics and Mathematical Sciences
Remark 2.4. In Theorem 2.3,
i if we choose h1 h2 1/2, r1 r2 6, and s 1 in 2.15, then we get
2
1 1
5
I f
, , 6, 6 ≤
Mb − ad − c,
2 2
72
2.20
ii if we choose h1 h2 1/2, r1 r2 2, and s 1 in 2.15, then we get
2
1 1
1
I f
, , 2, 2 ≤
Mb − ad − c,
2 2
8
2.21
∂2 f
∂2 f
∂2 f
∂2 f
M
a, c a, d b, c b, d,
∂t∂λ
∂t∂λ
∂t∂λ
∂t∂λ
2.22
where
which implies that Theorem 2.3 is a generalization of Theorem 1.6.
Theorem 2.5. Let f : Δ → R2 be a partial differentiable mapping on Δ a, b × c, d ⊂ R2 . If
∂2 f/∂t∂λ is bounded, that is,
∂2 f 2
let ∂ f
ta 1 − tb, λc 1 − λd
∂t∂λ ∂t∂λ
∞
∞
∂2 f
sup
ta 1 − tb, λc 1 − λd < ∞
∂t∂λ
x,y∈a,b×c,d
2.23
for all t, λ ∈ 0, 12 , then, for r1 , r2 ≥ 2 and h1 , h2 ∈ 0, 1 with 1/r1 ≤ h1 ≤ r1 − 1/r1 and
1/r2 ≤ h2 ≤ r2 − 1/r2 , the following inequality holds:
1
I f h1 , h2 , r1 , r2 b − ad − c
2 1
1
2 − r1 2 − r2 ∂f 2
2
≤
− h1 h1 − h2 h2 .
∂t∂λ 2
2
r12
r22
∞
2.24
Proof. From Lemma 2.1, using the property of modulus and the boundedness of ∂2 f/∂t∂s, we
get
1
I f h1 , h2 , r1 , r2 b − ad − c
∂2 f 1 ph1 , r1 , tqh2 , r2 , λ dtdλ.
≤
∂t∂λ 0
∞
2.25
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11
By the simple calculations, we have
i
ii
1
ph1 , r1 , tdt 1 − h1 h2 2 − r1 ,
1
2
r12
0
1
qh2 , r2 , λdt 1 − h2 h2 2 − r2 .
2
2
r22
0
2.26
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 h1 h2 1
and r1 r2 6, then we get
2
2 ∂2 f 1 1
5
I f
, , 6, 6 ≤
b − ad − c,
2 2
36 ∂t∂λ 2.28
∞
ii if we choose h1 h2 1/2 and r1 r2 2, then we get
2 2 1 1
1 ∂f I f
, , 2, 2 ≤
b − ad − c,
2 2
4 ∂t∂λ 2.29
∞
which implies that Theorem 2.5 is a generalization of Theorem 1.7.
The following theorem is a generalization of Theorem 1.6.
Theorem 2.7. Let f : Δ a, b × c, d ⊂ R2 → R2 be a partial differentiable mapping on
Δ a, b × c, d. If |∂2 f/∂t∂λ|q q > 1 is in L1 Δ and is a coordinated s-convex mapping in the
second sense on Δ, then, for r1 , r2 ≥ 2 and h1 , h2 ∈ 0, 1 with 1/r1 ≤ h1 ≤ r1 − 1/r1 and
1/r2 ≤ h2 ≤ r2 − 1/r2 , the following inequality holds:
1
I f h1 , h2 , r1 , r2 ≤ μ1/p ν1/p
3
3
b − ad − c
2
∂ f/∂t∂λa, cq ∂2 f/∂t∂λa, dq ∂2 f/∂t∂λb, cq ∂2 f/∂t∂λb, dq 1/p
×
,
s 12
2.30
12
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where
μ3 2 r1 − r1 h1 − 1p1 r1 h1 − 1p1
,
p1 r1 p 1
2 r2 − r2 h2 − 1p1 r2 h2 − 1p1
ν3 .
p1 r2 p 1
2.31
Proof. From Lemma 2.1, we can write
1
I f h1 , h2 , r1 , r2 b − ad − c
1
ph1 , r1 , tqh2 , r2 , λ
≤
0
∂2 f
×
ta 1 − tb, λc 1 − λddtdλ
∂t∂λ
1
1/p
p
≤
ph1 , r1 , tqh2 , r2 , λ dtdλ
2.32
0
q
1 1/q
∂2 f
×
.
ta 1 − tb, λc 1 − λd dtdλ
0 ∂t∂λ
Hence, by the inequality 2.32 and the coordinated s-convexity in the second sense of
|∂2 f/∂t∂λ|q , it follows that
1
I f h1 , h2 , r1 , r2 b − ad − c
1
1/p
p
ph1 , r1 , tqh2 , r2 , λ dtdλ
≤
×
0
|∂2 f/∂t∂λa, c|q |∂2 f/∂t∂λa, d|q |∂2 f/∂t∂λb, c|q |∂2 f/∂t∂λb, d|q
s 12
1/q
.
2.33
Note that
i
1
p1
r1 h1 − 1p1
ph1 , r1 , tp dt 2 r1 − r1 h1 − 1
,
p1 r1 p 1
2.34
p1
r2 h2 − 1p1
qh2 , r2 , λp dλ 2 r2 − r2 h2 − 1
.
p1 r2 p 1
2.35
0
ii
1
0
By the inequality 2.33 and the equalities 2.34 and 2.35, the assertion 2.30 holds.
International Journal of Mathematics and Mathematical Sciences
13
Remark 2.8. In Theorem 2.7,
i if we choose h1 h2 1/2, r1 r2 6, and s 1, then we get
2/p
1 1
2 1 2p1
1/q
I f
, , 6, 6 ≤
b − ad − cMq ,
p1
2 2
p1
6
2.36
ii if we choose h1 h2 1/2, r1 r2 2, and s 1, then we get
2/p
1 1
1
1/q
I f
, , 2, 2 ≤
b − ad − cMq ,
p
2 2
2 p1
2.37
iii if we choose h1 h2 1/2, r1 r2 6, s 1, and q 1, then we get
2
1 1
≤ 5
I f
,
,
6,
6
b − ad − cM1 ,
2 2
36
2.38
where
Mq 2
∂ f/∂t∂λa, cq ∂2 f/∂t∂λa, dq ∂2 f/∂t∂λb, cq ∂2 f/∂t∂λb, dq
4
. 2.39
Theorem 2.9. Let f : Δ → R2 be a partial differentiable mapping on Δ a, b × c, d ⊂ R2 . If
|∂2 f/∂t∂λ|q q ≥ 1 is in L1 Δ and is a coordinated s-convex mapping in the second sense on Δ, then,
for r1 , r2 ≥ 2 and h1 , h2 ∈ 0, 1 with 1/r1 ≤ h1 ≤ r1 − 1/r1 and 1/r2 ≤ h2 ≤ r2 − 1/r2 ,
the following inequality holds:
1
I f h1 , h2 , r1 , r2 b − ad − c
1−1/q
1
1
2 − r1 2 − r2 2
2
≤
− h1 h1 − h2 h2 2
2
r12
r22
∂2 f
∂2 f
q
q
× μ1 μ2 a, c ν2 a, d
∂t∂λ
∂t∂λ
q
q 1/q
∂2 f
∂2 f
ν1 μ2 ,
b, c ν2 b, d
∂t∂λ
∂t∂λ
where μi and νi i 1, 2 are as given in Theorem 2.3.
2.40
14
International Journal of Mathematics and Mathematical Sciences
Proof. From Lemma 2.1, we can write
1
I f h1 , h2 , r1 , r2 b − ad − c
1
ph1 , r1 , tqh2 , r2 , λ
≤
0
∂2 f
×
ta 1 − tb, λc 1 − λddtdλ
∂t∂λ
1−1/q
1
ph1 , r1 , tqh2 , r2 , λdtdλ
≤
2.41
0
×
1
ph1 , r1 , tqh2 , r2 , λ
0
q
1/q
∂2 f
×
.
ta 1 − tb, λc 1 − λd dtdλ
∂t∂λ
By the simple calculations, we have
i
ii
1
ph1 , r1 , tdt 1 − h1 h2 2 − r1 ,
1
2
r12
0
2.42
qh2 , r2 , λdλ 1 − h2 h2 2 − r2 .
2
2
r22
0
2.43
1
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,
q
∂2 f
ph1 , r1 , tqh2 , r2 , λ
ta 1 − tb, λc 1 − λd dtdλ
∂t∂λ
0
1
ph1 , r1 , tqh2 , r2 , λ
≤
0
q
q
2
2
s ∂ f
s s ∂ f
s
× tλ
a, c t 1 − λ a, d
∂t∂λ
∂t∂λ
q
q 2
2
s ∂ f
s s ∂ f
s
1 − t λ b, c 1 − t 1 − λ b, d dtdλ
∂t∂λ
∂t∂λ
1
International Journal of Mathematics and Mathematical Sciences
15
q
1
1
∂2 f
s
s
ph1 , r1 , t t dt
qh2 , r2 , λ λ dλ a, c
∂t∂λ
0
0
q 1
2
qh2 , r2 , λ1 − λs dλ ∂ f a, d
∂t∂λ
0
q
1
1
∂2 f
s
s
ph1 , r1 , t1 − t dt
qh2 , r2 , sλ dλ b, c
∂t∂s
0
0
q 1
∂2 f
s
qh2 , r2 , λ1 − λ dλ b, d
∂t∂λ
0
q
q ∂2 f
∂2 f
μ1 μ 2 a, c ν2 a, d
∂t∂λ
∂t∂λ
q
q ∂2 f
∂2 f
2.44
ν1 μ2 b, c ν2 b, d .
∂t∂λ
∂t∂λ
By 2.41–2.44, the assertion 2.40 holds.
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. Özdemir, A. O. Akdemir, H. Kavurmaci, and M. Avci, “On the Simpson’s inequality for coordinated 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 Hadamardtype 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-ordinated s, m-convex mappings in the second
sense,” Far East Journal of Mathematical Sciences, vol. 51, no. 2, pp. 205–216, 2011.
13 D.-Y. Hwang, K.-L. Tseng, and G.-S. Yang, “Some Hadamard’s inequalities for co-ordinated convex
functions in a rectangle from the plane,” Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 63–73,
2007.
16
International Journal of Mathematics and Mathematical Sciences
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.
15 M. A. Latif and M. Alomari, “Hadamard-type inequalities for product two convex functions on the
co-ordinates,” International Mathematical Forum, vol. 4, no. 45-48, pp. 2327–2338, 2009.
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