Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 493531, 13 pages
doi:10.1155/2011/493531
Research Article
Generalization of Some Simpson-Like Type
Inequalities via Differentiable s-Convex Mappings
in the Second Sense
Jaekeun Park
Department of Mathematics, Hanseo University, Seosan-Si, Chungnam-Do 356-706, Republic of Korea
Correspondence should be addressed to Jaekeun Park, jkpark@hanseo.ac.kr
Received 31 March 2011; Accepted 24 June 2011
Academic Editor: Jewgeni Dshalalow
Copyright q 2011 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.
The author obtained new generalizations and refinements of some inequalities based on
differentiable s-convex mappings in the second sense. Also, some applications to special means
of real numbers are given.
1. Introduction
Recall that the function f : 0, b → R, b > 0 is said to be s-convex in the second sense for
s ∈ 0, 1 if
f tx 1 − ty ≤ ts fx 1 − ts f y ,
1.1
for all x, y ∈ 0, b and t ∈ 0, 1 1–5.
In 1.1, if we let s 1, f : I ⊂ 0, ∞ → R is said to be a convex mapping on an interval
I 1.
Let us denote the set of s-convex mappings in the second sense on I by Ks2 I.
For some further properties of the s-convex mappings, see 1–3, 6. In recent, M. Z.
Sarikaya et al. 4, and U. S. Kirmaci et al. 7 established a more general result of the HermiteHadamard inequalities.
For recent years many authors have established error estimations for the Simpson’s
inequality: for refinements, counterparts, generalizations, and new Simpson’s type inequalities, see 1, 4, 6, 8.
2
International Journal of Mathematics and Mathematical Sciences
S. S. Dragomir et al. 9, and M. Alomari et al. 8 proved the following developments
on Simpson’s inequality for which the remainder is expressed in terms of lower derivatives
than the twice.
In the sequel, denote the interior of an interval I by I0 .
Theorem 1.1. Let f : I ⊂ 0, ∞ → R be an absolutely continuous mapping on a, b such that
f ∈ Lp a, b, where a, b ∈ I with a < b. Then the following inequality holds:
b
1 fa fb
b − a1/q 2q1 1 1/q ab
1
f . 1.2
2f
−
fxdx ≤
p
3
2
2
b−a a
6
3 q1
In this article, the author gives some generalized Simpson’s type inequalities based on
s-convex mappings in the second sense by using the following lemma.
2. Generalization of Inequalities Based on s-Convex Mappings
In this article, for the simplicity of the notation, let
1
1
Sba f h, n fa n − 2fhb 1 − ha fb −
n
b−a
b
fxdx,
2.1
a
for h ∈ 0, 1 with 1/n ≤ h ≤ n − 1/n for any integer n ≥ 2.
In order to generalize the classical Simpson-like type inequalities, we need the
following lemma 1, 6.
Lemma 2.1. Let f : 0, b → R, b > 0 be a differentiable mapping on I0 such that f ∈ La, b,
where a, b ∈ I with a < b and a, b ⊂ 0, b. If f ∈ L1 a, b, then, for h ∈ 0, 1 with 1/n ≤ h ≤
n − 1/n for any n ≥ 2 the following equality holds:
Sba
f h, n b − a
1
pt, hf tb 1 − tadt,
2.2
0
for each t ∈ 0, 1, where
⎧
1
⎪
⎨t − ,
t ∈ 0, h,
n
pt, h ⎪
⎩t − n − 1 , t ∈ h, 1.
n
2.3
International Journal of Mathematics and Mathematical Sciences
3
Proof. By the integration by parts, we have
h
1
1
n−1 f tb 1 − tadt f tb 1 − tadt
t−
t−
n
n
0
h
b
1
1
n−2
1
fhb 1 − ha fxdx,
fa fb −
b−a
n
n
b − a2 a
2.4
which completes the proof.
Theorem 2.2. Let f : 0, b → R, b > 0 be a differentiable mapping on I0 such that f ∈ La, b,
where a, b ∈ I with a < b and a, b ⊂ 0, b. If |f | ∈ Ks2 a, b, for some s ∈ 0, 1, then for
h ∈ 0, 1 with 1/n ≤ h ≤ n − 1/n for any n ≥ 2 the following inequality holds:
b
Sa f h, n ≤ b − a λ11 f b μ11 f a ,
2.5
where
2s−n
2 2n − 1s2
s2
n s 1s 2 ns 1s 2
λ11 μ11
hs1 s2h − 1 2h − 1
,
s 1s 2
2s−n
2 2n − 1s2
s2
n s 1s 2 ns 1s 2
2.6
1 − hs1 {s1 − 2h − 2h}
.
s 1s 2
Proof. From Lemma 2.1 and since |f | is s-convex on a, b, by using Hölder integral inequality,
we have
b
Sa f h, n
1/n 1
− t | ts f b 1 − ts f a dt
n
0
h 1 s t f b 1 − ts f a dt
t−
b − a
n
1/n
n−1/n n−1
− t ts f b 1 − ts f a dt
b − a
n
h
1
n − 1 s t f b 1 − ts f a dt
t−
b − a
n
n−1/n
b − a λ11 f b μ11 f a ,
≤ b − a
which implies the theorem.
2.7
4
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Corollary 2.3. In Theorem 2.2, one has:
i
b 1
2 2n−1s2 ns1 s−n2−2−s1 n Sa f
f a f b ,
, n ≤ b − a
2
ns2 s1s2
2.8
ii
−s b 1
6 15s2 −3s2 3s−4 f a f b ,
Sa f
, 6 ≤ b − a
2
18s1s2
2.9
which implies that Corollary 2.3 is a generalization of Theorem 1.1.
Theorem 2.4. Let f : 0, b → R, b > 0 be a differentiable mapping on I0 such that f ∈ La, b,
where a, b ∈ I with a < b and a, b ⊂ 0, b. If |f |q ∈ Ks2 a, b, for some fixed s ∈ 0, 1 and q > 1
with 1/p 1/q 1, then for h ∈ 0, 1 with 1/n ≤ h ≤ n − 1/n for any n ≥ 2 the following
inequality holds:
1/p 1/q
1 hn − 1p1
h
b
Sa f h, n ≤ b − a
s1
np1 p 1
q q 1/q
× f hb 1 − ha f a
b − a
1 n − hn − 1
np1 p 1
p1
1/p 1−h
s1
1/q
2.10
q q 1/q
× f b f hb 1 − ha
.
Proof. From Lemma 2.1, using the Hölder inequality we get
h 1/q
1/p h
q
1 p
b
f tb 1 − ta dt
Sa f h, n ≤ b − a
t − n dt
0
0
1 1/q
1/p 1
q
n − 1 p
ftb 1 − ta dt
b − a
.
t − n dt
h
h
2.11
Since |f |q ∈ Ks2 a, b for a fixed s ∈ 0, 1, we have
a
h
0
f tb 1 − taq dt ≤
h
f hb 1 − haq f aq ,
s1
2.12
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5
b
1
f tb 1 − taq dt ≤
h
q 1 − h q f b f hb 1 − ha .
s1
2.13
By 2.11 and 2.12, the assertion 2.10 holds.
Corollary 2.5. In Theorem 2.4, letting n 6 and h 1/2, one has
1/p 1/q
b 1
2p1 1
1
S a f
, 6 ≤ b − a
2
2s 1
6p1 p 1
2.14
a b q q 1/q
q a b q 1/q
f b
f a f
.
×
f
2
2
Theorem 2.6. Let f : 0, b → R, b > 0 be a differentiable mapping on I0 such that f ∈ La, b,
where a, b ∈ I with a < b and a, b ⊂ 0, b. If |f |q ∈ Ks2 a, b, for some fixed s ∈ 0, 1 and
q > 1 with 1/p 1/q 1, then for h ∈ 0, 1 with 1/n ≤ h ≤ n − 1/n for any n ≥ 2 the following
inequality holds:
b
Sa f h, n ≤ b − a
×
1
np1 p 1
1 hn − 1p1
1/p 1
s1
1/q
1/p q q 1/q
hs1 f b 1 − 1 − hs1 f a
1/p q
q 1/q
1 n − nh − 1p1
×
1 − hs1 f b 1 − hs1 f a
.
2.15
Proof. Note that
1 − 1 − hs1 q
f a ,
a
s1
0
1
q
1 − hs1 q
1 − hs1 q
f tb 1 − ta dt ≤
f b f a .
b
s1
s1
h
h
f tb 1 − taq dt ≤
hs1
s1
q
f b By 2.11 and 2.16, the assertion 2.15 of this theorem holds.
2.16
6
International Journal of Mathematics and Mathematical Sciences
Corollary 2.7. In Theorem 2.6, letting h 1/2, then one has
1/p 1/q
b 1
2p1 n − 2p1
1
S a f
, n ≤ b − a
2
2s1 s 1
2p1 np1 p 1
×
1/q s1
q q 1/q
q s1
f b 2 − 1 f aq
2 − 1 f b f a
,
2.17
which implies that Theorem 2.6 is a generalization of Theorem 1.1.
Theorem 2.8. Let f : 0, b → R, b > 0 be a differentiable mapping on I0 such that f ∈ La, b,
where a, b ∈ I with a < b and a, b ⊂ 0, b. If |f |q ∈ Ks2 a, b, for some fixed s ∈ 0, 1 and
q ≥ 1 with 1/p 1/q 1, then for h ∈ 0, 1 with 1/n ≤ h ≤ n − 1/n for any n ≥ 2 the following
inequality holds:
1/p
q 1/q
q
1 nh − 12
b
λ21 f b ν21 f a
Sa f h, n ≤ b − a
2n2
b − a
1 nh − n 1
2n2
2
1/p
q 1/q
q
,
λ22 f b ν22 f a
2.18
where
hs1 −2 − s hn hns
2
,
ns 1s 2
ns2 s 1s 2
s1
s2
1
−
h
2
−
n
1
s
h1 − hs1
2n − 1
ν21 s2
−
,
ns 1s 2
s2
n s 1s 2
s 2 − n 1 hs1
hs1 1 − h
2n − 1s2
λ22 s2
−
,
ns 1s 2
s2
n s 1s 2
λ21 ν22 2.19
2
1 − hs1 −2 − s n − hn ns − hns
.
ns 1s 2
ns2 s 1s 2
Proof. Suppose that q ≥ 1. From Lemma 2.1, using the power mean inequality one has
⎡
1/q
h
1/p h q
1
1
b
Sa f h, n ≤ b − a⎣
t − n dt
t − n f tb 1 − ta dt
0
0
1 1/q ⎤
1/p 1 q
n
−
1
n
−
1
dt
f tb 1 − ta dt
t −
t −
⎦.
n n h
h
2.20
International Journal of Mathematics and Mathematical Sciences
7
Since |f | is s-convex on a, b, we have
h
t − 1 f tb 1 − taq dt ≤ λ21 f bq ν21 f aq
n
0
1
t − n − 1 ftb 1 − taq dt ≤ λ22 f bq ν22 f aq .
n
h
2.21
By the above facts 2.20 and 2.21, the assertion 2.18 in this theorem is proved.
Corollary 2.9. In Theorem 2.8, letting h 1/2, one has
λ21 ν22 λ22 ν21
2
n − 2s 1 − 2
,
ns2 s 1s 2 2s2 ns 1s 2
ns2 2−s2 s 1 2n − 1s2 s − n 2 2s1 1
s1
,
ns2 s 1s 2
2 ns 1s 2
2.22
which implies that
1/p
b 1
S a f
≤ b − a 1 − 1 1
,
n
2
8 2n n2
! q
q 1/q q
q 1/q "
.
× λ21 f b λ22 f a
λ22 f b λ21 f a
2.23
Especially, in Theorem 2.8, letting h 1/2 and m 1, one has
b 1
1
1
1 1/p
Sa f
, n ≤ b − a
−
2
8 2n n2
q
q 1/q
× λ21 f b λ22 f a
2.24
q
q 1/q .
λ22 f b λ21 f a
3. Applications to Special Means
We now consider the applications of our theorems to the followings special means.
a The arithmetic mean: Aa, b a b/2, a, b ≥ 0.
b The p-logarithmic mean:
⎧
1/p
⎪
p1
p1
⎪
⎨ b − a
, if a / b,
Lp a, b p 1 b − a
⎪
⎪
⎩a,
if a b,
3.1
8
International Journal of Mathematics and Mathematical Sciences
for p ∈ R \ {−1, 0} and a, b > 0.
Now, using the results of Section 2, some new inequalities are derived for the following
means:
1.1 Let f : a, b → R, 0 < a < b, fx xs , s ∈ 0, 1.
a In Theorem 2.2,
i if h 1/2 and n ≥ 2, then we get
1
2Aas , bs n − 2As a, b − Lss a, b
n
2 2n − 1s2 ns1 2 s − n − 2−s1 n
s−1 s−1
≤ b − a2s
,
A
a
,
b
ns2 s 1s 2
3.2
and,
ii if h 1/2 and n 6, then we have
−s
s2 −s
−s
1
6
5
6
3s
−
4
−
2
3
s
s
s
s
Aa , b 2A a, b − Ls a, b ≤ b − as
A as−1 , bs−1 .
3
9s 1s 2
3.3
b In Theorem 2.4,
i if h 1/2, n ≥ 2 and q > 1 then we get:
1
2Aas , bs n − 2As a, b − Lss a, b
n
1/p 1/q
b − a 1 n/2 − 1p1
sq
≤
2
2s 1
np1 p 1
1/q 1/q s−1q
s−1q
s−1q
s−1q
× A
A
,
Aa, b a
Aa, b b
3.4
and
ii if h 1/2, n 6 and q > 1, then we have
1
Aas , bs 2As a, b − Lss a, b
3
1/p 1/q
b − a 1 2p1
sq
≤
12
s 1
3 p1
1/q 1/q s−1q
s−1q
s−1q
s−1q
× A
A
,
Aa, b a
Aa, b b
c In Theorem 2.6,
3.5
International Journal of Mathematics and Mathematical Sciences
9
i if h 1/2, n ≥ 2 and q > 1 then we get
1
2Aas , bs n − 2As a, b − Lss a, b
n
1/p 1/q
1 n/2 − 1p1
sq
≤ b − a
2s1 s 1
np1 p 1
×
3.6
1/q 1/q bs−1q 2s1 − 1 as−1q
2s1 − 1 bs−1q as−1q
,
and
ii if h 1/2, n 6 and q > 1, then we have
1
Aas , bs 2As a, b − Lss a, b
3
1/p 1/q
1 2p1
sq
≤ b − a p1 2s1 s 1
6
p1
×
3.7
1/q 1/q bs−1q 2s1 − 1 as−1q
2s1 − 1 bs−1q as−1q
.
In Theorem 2.8,
i if h 1/2, n ≥ 2 and q ≥ 1 then we get
1
2Aas , bs n − 2As a, b − Lss a, b
n
1/p
1 n/2 − 12
≤ b − a
2n2
×
3.8
1/q 1/q λ21 bs−1q λ22 as−1q
λ22 bs−1q λ21 bs−1q
,
where
λ21 λ22
2
ns/2 n/2 − s − 2
,
ns2 s 1s 2
n2s1 s 1s 2
s − n 2 2s1 1
2n − 1s2
1
,
s2
− s2
n s 1s 2
2 s 2
n2s1 s 1s 2
3.9
10
International Journal of Mathematics and Mathematical Sciences
and
ii if h 1/2, n 6 and q ≥ 1, then we have
1
{Aas , bs 2As a, b} − Lss a, b
3
1/p 1/q 1/q 5
s−1q
s−1q
s−1q
s−1q
≤ b − a
λ22 a
λ22 b
λ21 b
λ21 b
,
72
3.10
where
2 3s1 2s 1
,
6s2 s 1s 2
2 · 5s2 3s1 2s1 − 2 s − 3s1 2s3 7
.
6s2 s 1s 2
λ21 λ22
3.11
2.2 Let f : a, b → R, 0 < a < b, fx 1/xs , s ∈ 0, 1.
a In Theorem 2.2,
i if h 1/2 and n ≥ 2, then we get
1 # −s −s $
−s
−s
2A a , b
n − 2A a, b − L−s a, b
n
⎧
⎫
⎨ 2s2 1 n − 1s2 2s1 ns1 2 s − n − ns2 ⎬
≤ b − a2s
⎩
⎭
ns2 2s1 s 1s 2
3.12
× A a−s1 , b−s1 ,
and
ii if h 1/2 and n 6, then we have
1 # −s −s $
−s
−s
A a ,b
2A a, b − L−s a, b
3
2
1 5s2 3s − 46s − 3s2
−s1 −s1
≤ b − as
.
A
a
,
b
3
6s2 s 1s 2
b In Theorem 2.4,
3.13
International Journal of Mathematics and Mathematical Sciences
11
i if h 1/2 and n ≥ 2, then we get
1 # −s −s $
−s
−s
2A a , b
n − 2A a, b − L−s a, b
n
1/p 1/q
1 n/2 − 1p1
sq
≤ b − a
2s 1
np1 p 1
1/q 1/q −s1q
−s1q
−s1q
−s1q
× A
A
,
a, b a
a, b b
3.14
and
ii if h 1/2 and n 6, then we have
1 # −s −s $
−s
−s
A a ,b
b
−
L
b
2A
a,
a,
−s
3
1/p 1/q
1 2p1
sq
≤ b − a p1 s 1
6
p1
3.15
× A1/q A−s1q a, b, a−s1q A1/q A−s1q a, b, b−s1q .
c In Theorem 2.6,
i if h 1/2 and n ≥ 2, then we get
1 # −s −s $
−s
−s
2A a , b
b
−
L
b
−
2A
a,
a,
n
−s
n
1/p 1/q p1 1/p
n
1
sq
≤ b − a p1 −
1
1
2
2s1 s 1
n
p1
3.16
1/q 1/q −s1q
s1
−s1q
s1
−s1q
−s1q
× b
2 −1 a
2 −1 b
a
,
and
ii if h 1/2 and n 6, then we have
1 # −s −s $
−s
−s
A a ,b
b
−
L
b
2A
a,
a,
−s
3
1/p 1/q
1 2p1
sq
≤ b − a p1 2s1 s 1
6
p1
1/q 1/q −s1q
s1
−s1q
s1
−s1q
−s1q
× b
2 −1 a
2 −1 b
a
.
3.17
12
International Journal of Mathematics and Mathematical Sciences
d In Theorem 2.8,
i if h 1/2, n ≥ 2 and q ≥ 1 then we get
1 # −s −s $
−s
−s
−s
2A a , b
−
2A
−
ha
−
L
b
2
n
hb,
1
a,
−s
n
1/p
n − 22 4
≤ b − as
8n2
3.18
1/q 1/q s−1q
s−1q
s−1q
s−1q
× λ31 b
λ32 a
λ32 b
λ31 b
,
where
λ31 λ32
ns2 s
−2 − s n/2 ns/2
2
,
n2s2 s 1s 2
1s 2
s − n 2 2s1 1
2n − 1s2
1
,
s2
− s2
s1
n s 1s 2
2 s 2
n2 s 1s 2
3.19
and
ii if h 1/2, n 6 and q ≥ 1, then we have
1 # −s −s $
−s
−s
A a ,b
b
−
L
b
2A
a,
a,
−s
3
1/p 1/q 1/q 3.20
5
qs−1
qs−1
qs−1
qs−1
≤ b − as
× λ31 b
λ32 a
λ32 b
λ31 b
,
72
where
λ31 λ32
12 3s2 2s 1
,
6s3 s 1s 2
2 · 5s2 3s1 2s1 − 4 s 3s1 1 − 2s3
.
6s2 s 1s 2
3.21
Acknowledgments
The author is thankful to Professor Merve Avci and Uğur S. Kirmaci, Atatürk University, K.
K. Education Faculty, Deptartment of Mathematics, for their valuable suggestions and for the
improvement of this paper. This work is financially supported by Hanseo University research
fund no. 111-S-101.
International Journal of Mathematics and Mathematical Sciences
13
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Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014