Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 560586, 14 pages
doi:10.1155/2012/560586
Research Article
On Integral Inequalities of Hermite-Hadamard
Type for s-Geometrically Convex Functions
Tian-Yu Zhang,1 Ai-Ping Ji,1 and Feng Qi2
1
College of Mathematics, Inner Mongolia University for Nationalities,
Inner Mongolia Autonomous Region, Tongliao City 028043, China
2
School of Mathematics and Informatics, Henan Polytechnic University,
Jiaozuo City, Henan Province, 454010, China
Correspondence should be addressed to Feng Qi, qifeng618@gmail.com
Received 12 May 2012; Accepted 19 May 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Tian-Yu Zhang et al. 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 authors introduce the concept of the s-geometrically convex functions. By the well-known
Hölder inequality, they establish some integral inequalities of Hermite-Hadamard type related to
the s-geometrically convex functions and apply these inequalities to special means.
1. Introduction
We firstly list several definitions and some known results.
Definition 1.1. A function f : I ⊂ R −∞, ∞ → R is said to be convex if
f λx 1 − λy ≤ λfx 1 − λf y
1.1
for x, y ∈ I and λ ∈ 0, 1.
Definition 1.2 see 1. A function f : I ⊂ R0 0, ∞ → R0 is said to be s-convex if
f λx 1 − λy ≤ λs fx 1 − λs f y
for some s ∈ 0, 1, where x, y ∈ I, and λ ∈ 0, 1.
If s 1, the s-convex function becomes a convex function on R0 .
1.2
2
Abstract and Applied Analysis
Theorem 1.3 2, Theorem 2.2. Letf : I ◦ ⊂ R → R be a differentiable mapping on I ◦ , a, b ∈ I ◦ ,
a < b.
i If |f x| is a convex function on a, b, then
b
b − af a f b
fa fb
1
−
.
fxdx ≤
2
b−a a
8
1.3
ii If |f x|p/p−1 is a convex function on a, b, for p > 1, then
p/p−1 p/p−1 b
fa fb
f a
f b
b−a
1
−
fxdx ≤ 1/p
2 p1
2
b−a a
2
p−1/p
.
1.4
Theorem 1.4 3, Theorems 2.3 and 2.4. Letf : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I,
a < b. If |f x|p is convex on a, b, for p > 1, then
b
b − a 4 1/p a b
1
f a f b ,
−
fxdx ≤
f
2
b−a a
4
p1
b
b − a 4 1/p
a b
1
−
fxdx ≤
f
2
b−a a
16
p1
p/p−1
p/p−1 p−1/p
× f a
3f b
1.5
p/p−1 p/p−1 p−1/p
3 f a
f b
.
Theorem 1.5 4, Theorem 1–4. Letf : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I, a < b. If
|f x|q is s-convex on a, b, for q > 1, then
b
fa fb
1
−
fxdx
2
b−a a
1/q
s s 1/q
b − a 1 q−1/q
2 1/2s
f a f b
≤
.
2
2
s 1s 2
1.6
Abstract and Applied Analysis
3
Theorem 1.6 5, Theorem 4. Letf : I → R0 be differentiable on I ◦ , a, b ∈ I, a < b, and f ∈
La, b. If |f x|q is s-convex on a, b for some s ∈ 0, 1, and p, q ≥ 1, such that 1/q1/p 1
then
q
b
a b
f a s 1f a b/2q 1/q
1
−1/p
−
fxdx ≤ 2
f
2
b−a a
{s 1s 2}1/q
−1/p
2
−1/p
2
q
f b s 1f a b/2q 1/q
{s 1s 2}1/q
a b q 1/q
q
βs 1, 2 f a βs 2, 1f
2
q
a b q 1/q
.
βs 1, 2 f b βs 2, 1f
2
1.7
Theorem 1.7 6, Theorems 2.2–2.4. Letf : I → R0 be differentiable on I ◦ , a, b ∈ I, a < b, and
f ∈ La, b.
i If |f x| is s-convex on a, b for some s ∈ 0, 1, then
b
a b
ab b−a
1
f
≤
f
f
f
2s
1
−
fxdx
a
b
4s 1s 2
2
b−a a
2
2−s
2 1 b − a f a f b .
≤
4s 1s 2
1.8
ii If |f x|p/p−1 p > 1 is a s-convex function on a, b for some s ∈ 0, 1, then
b
a b
1
−
fxdx
f
2
b−a a
≤
1/p 2/q q
q 1/q
b−a
1
1
21−s s 1 f a 21−s f b
4
p1
s1
q p/p−1 1−s
q 1/q
21−s f a f a
2 s 1 f b
,
1.9
where 1/p 1/q 1.
4
Abstract and Applied Analysis
iii If |f x|q q ≥ 1 is s-convex on a, b for some s ∈ 0, 1, then
b
a b
1
−
fxdx
f
2
b−a a
≤
1/q q
q 1/q
b−a
2
21−s 1 f a 21−s f b
8
s 1s 2
q q q 1/q
21−s f a f a 21−s 1 f b
.
1.10
Now we introduce the definition of the s-geometrically convex function.
Definition 1.8. A function f : I ⊂ R 0, ∞ → R is said to be a geometrically convex
function if
λ 1−λ
f xλ y1−λ ≤ fx f y
1.11
for x, y ∈ I and λ ∈ 0, 1.
Definition 1.9. A function f : I ⊂ R → R is said to be a s-geometrically convex function if
λs 1−λs
f y
f xλ y1−λ ≤ fx
1.12
for some s ∈ 0, 1, where x, y ∈ I and λ ∈ 0, 1.
If s 1, the s-geometrically convex function becomes a geometrically convex function
on R .
In this paper, we will establish some integral inequalities of Hermite-Hadamard type
related to the s-geometrically convex functions and then apply these inequalities to special
means.
2. A Lemma
In order to prove our results, we need the following lemma.
Lemma 2.1. Letf : I ⊂ R → R be differentiable on I ◦ , and a, b ∈ I, with a < b. If f ∈ La, b,
then
f
b
1
fxdx
b−a a
ab
ab
b−a 1
t − 1f 1 − t
tb dt,
tf 1 − ta t
4
2
2
0
ab
2
−
Abstract and Applied Analysis
5
b
fa fb
1
−
fxdx
2
b−a a
ab
b−a 1
ab
tf 1 − t
tb dt.
t − 1f 1 − ta t
4
2
2
0
2.1
Proof. Integrating by part and changing variables of integration yields
1
0
1
0
ab
ab
t − 1f 1 − t
tb dt
tf 1 − ta t
2
2
1 2
a b 1
ab
tf 1 − ta t
dt
− f 1 − ta t 2
b−a
2
0
0
1 1 a
b
ab
2
tb − f 1 − t
tb dt
t − 1f 1 − t
b−a
2
2
0
0
b
ab
4
1
f
fxdx ,
−
b−a
2
b−a a
ab
ab
tf 1 − t
tb dt
t − 1f 1 − ta t
2
2
1 2
ab
a b 1
dt
t − 1f 1 − ta t
− f 1 − ta t 2
b−a
2
0
0
1 1 ab
ab
2
tb − f 1 − t
tb dt
tf 1 − t
b−a
2
2
0
0
b
fa fb
1
4
−
fxdx .
b−a
2
b−a a
2.2
This completes the proof of Lemma 2.1.
3. Main Results
Theorem 3.1. Letf : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I, with a < b, and f ∈ La, b.
If |f x|q is s-geometrically convex and monotonically decreasing on a, b for q ≥ 1 and s ∈ 0, 1,
then
b
b − a 1 1−1/q a b
1
−
fxdx ≤
G1 s, q; g1 α, g2 α
f
2
b−a a
4
2
b
b − a 1 1−1/q fa fb
1
−
fxdx ≤
G1 s, q; g2 α, g1 α ,
2
b−a a
4
2
3.1
3.2
6
Abstract and Applied Analysis
where
⎧
1
⎪
⎪
α 1,
⎨ ,
2
g1 α α
ln α − α 1
⎪
⎪
, α
/ 1,
⎩
ln α2
⎧
1
⎪
⎪
⎨ ,
g2 α 2α − ln α − 1
⎪
⎪
⎩
ln α2
α 1,
α/
1,
−u v
αu, v f a f b , u, v > 0,
3.3
3.4
G1 s, q; g1 α, g2 α
⎧
s sq sq 1/q s/2 sq sq 1/q
⎪
⎪
,
,
g2 α
f af b
,
⎪f a g1 α
⎪
2 2
2 2 ⎪
⎪
⎪
⎪
f a ≤ 1,
⎪
⎪
⎪
⎪
⎪
1/q
1/q
⎪
⎪
1/2s s/2
q sq
⎪
⎨f a1/s g1 α q , sq
f b
,
f a
,
g2 α
2s 2
2s 2 ⎪
f b ≤ 1 ≤ f a,
⎪
⎪
⎪
⎪
⎪
⎪
1/q
1/q
⎪
⎪
1/s
1/2s
q q
q q
⎪
⎪
⎪f a
,
,
f af b
,
g1 α
g2 α
⎪
⎪
2s 2s
2s 2s ⎪
⎪
⎩
1 ≤ f b.
3.5
Proof. 1 Since |f x|q is s-geometrically convex and monotonically decreasing on a, b,
from Lemma 2.1 and Hölder inequality, we have
b
ab
1
−
fxdx
f
2
b−a a
1
a b f 1 − t a b tb dt
−
1|
tf 1 − ta t
|t
2
2
0
⎧
1 1−1/q 1 1/q
b − a⎨
a b q
≤
tdt
tf 1 − ta t
dt
4 ⎩ 0
2
0
≤
b−a
4
1
1−1/q 1
1 − tdt
0
0
⎫
q 1/q ⎬
a
b
tb dt
1 − tf 1 − t
⎭
2
⎧
1/q
1−1/q ⎨ 1 b−a 1
2−t/2 t/2 q
tf a
b
≤
dt
⎩ 0
4
2
1
0
1/q ⎫
⎬
q
1 − tf a1−t/2 b1t/2 dt
⎭
Abstract and Applied Analysis
⎧
1/q
1−1/q ⎨ 1
q2−t/2s qt/2s
b−a 1
f b
≤
tf a
dt
⎩ 0
4
2
1
7
1/q ⎫
⎬
q1−t/2s q1t/2s
f b
dt
1 − tf a
0
.
⎭
3.6
If 0 < μ ≤ 1 ≤ η, 0 < α, s ≤ 1, then
s
μα ≤ μαs ,
s
ηα ≤ ηα/s .
3.7
i If |f a| ≤ 1, by 3.7, we obtain that
1 q2−t/2s qt/2s f b
dt
t f a
0
1
1 sq/22−t sqt/2 sq sq sq f b
≤
dt f a g1 α
,
,
t f a
2 2
0
q1−t/2s q1t/2s f b
dt
1 − t f a
0
≤
3.8
1
sq/21−t sq/21t sq/2 sq sq f b
,
.
dt f af b
g2 α
1 − t f a
2 2
0
ii If |f b| ≤ 1 ≤ |f a|, by 3.7, we obtain that
1 q2−t/2s qt/2s f b
dt
t f a
0
1
1 q/s
q/2s2−t sq/2t q sq
f b
dt f a
≤
,
,
t f a
g1 α
2s 2
0
q1−t/2s q1t/2s f b
dt
1 − t f a
0
≤
1
q/2s1−t sq/21t q/2s sq/2
q sq
f b
f b
,
dt f a
g2 α
.
1 − t f a
2s 2
0
3.9
iii If 1 ≤ |f b|, by 3.7, we obtain that
1 q2−t/2s qt/2s f b
dt
t f a
0
1 q/s
q/2s2−t q/2st q q
f b
,
,
≤
dt f a
t f a
g1 α
2s 2s
0
8
Abstract and Applied Analysis
1
q1−t/2s q1t/2s f b
dt
1 − t f a
0
≤
1
q/2s1−t q/2s1t q/2s
q q
f b
,
.
dt f af b
g2 α
1 − t f a
2s 2s
0
3.10
From 3.6 to 3.10, 3.1 holds.
2 Since |f x|q is s-geometrically convex and monotonically decreasing on a, b,
from Lemma 2.1 and Hölder inequality, we have
b
fa fb
1
−
fxdx
2
b−a a
⎡
1−1/q 1
1
1/q
b − a⎣
a b q
≤
1 − tdt
1 − tf 1 − ta t
dt
4
2
0
0
1
0
⎤
1−1/q 1 q 1/q
a
b
⎦
tb dt
tdt
tf 1 − t
2
0
3.11
⎧
1/q
q2−t/2s qt/2s
b − a 1 1−1/q ⎨ 1
f b
dt
≤
1 − t f a
⎩ 0
4
2
1
1/q ⎫
⎬
q1−t/2s q1t/2s
f b
tf a
dt
0
.
⎭
i If |f a| ≤ 1, by 3.7, we have
1
q2−t/2s qt/2s sq sq sq f b
dt ≤ f a g2 α
,
,
1 − t f a
2 2
0
1 q1−t/2s q1t/2s sq/2 sq sq f b
,
.
dt ≤ f af b
t f a
g1 α
2 2
0
3.12
ii If |f b| ≤ 1 ≤ |f a|, by 3.7, we have
1
q2−t/2s qt/2s q/s
q sq
f b
dt ≤ f a g2 α
,
,
1 − t f a
2s 2
0
1 q/2s sq/2
q1−t/2s q1t/2s q sq
f b
f b
,
.
dt ≤ f a
t f a
g1 α
2s 2
0
3.13
Abstract and Applied Analysis
9
iii If 1 ≤ |f b|, by 3.7, we have
1
q2−t/2s qt/2s q/s
q q
f b
dt ≤ f a g2 α
,
,
1 − t f a
2s 2s
0
1 q1−t/2s q1t/2s q/2s
q q
f b
,
.
dt ≤ f af b
t f a
g1 α
2s 2s
3.14
0
From 3.11 to 3.14, 3.2 holds. This completes the required proof.
Applying Theorem 3.1 to q 1, s 1, respectively, results in the following corollary.
Corollary 3.2. Letf : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I with a < b, and f ∈ La, b.
If |f x|q is s-geometrically convex and monotonically decreasing on a, b for s ∈ 0, 1, then
i when q 1, one has
b
b−a a b
1
G1 s, 1; g1 α, g2 α ,
fxdx ≤
−
f
2
b−a a
4
b
b−a fa fb
1
−
G1 s, 1; g2 α, g1 α .
fxdx ≤
2
b−a a
4
3.15
ii when s 1, one has
b
b − a 1 1−1/q a b
1
−
fxdx ≤
G1 1, q; g1 α, g2 α ,
f
2
b−a a
4
2
b
b − a 1 1−1/q fa fb
1
−
fxdx ≤
G1 1, q; g2 α, g1 α ,
2
b−a a
4
2
3.16
where g1 α, g2 α, αu, v, G1 s, q; g2 α, g1 α are same with 3.3–3.5.
Theorem 3.3. Let f : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I, with a < b, and f ∈
La, b. If |f x|q is s-geometrically convex and monotonically decreasing on a, b, for q > 1 and
s ∈ 0, 1, then
b
b − a q − 1 1−1/q a b
1
−
fxdx ≤
G2 s, q; g3 α ,
f
2
b−a a
4
2q − 1
b
b − a q − 1 1−1/q fa fb
1
−
fxdx ≤
G2 s, q; g3 α ,
2
b−a a
4
2q − 1
3.17
3.18
10
Abstract and Applied Analysis
where
G2 s, q; g3 α ⎧
1/q
⎪
f as f af bs/2 g3 α sq , sq
⎪
,
⎪
⎪
2 2
⎪
⎪
1/q
⎪
⎨ f a1/s f a1/2s f bs/2 g3 α q , sq
,
2s 2
⎪
⎪
⎪
⎪
q q 1/q
⎪
⎪
⎪
⎩ f a1/s f af b1/2s g3 α
,
,
2s 2s
⎧
⎨1,
α 1,
g3 α α − 1
⎩
, α/
1,
ln α
f a ≤ 1,
f b ≤ 1 ≤ f a,
1 ≤ f b,
3.19
and αu, v is the same as in 3.4.
Proof. 1 Since |f x|q is s-geometrically convex and monotonically decreasing on a, b,
from Lemma 2.1 and Hölder inequality, we have
b
a b
1
−
fxdx
f
2
b−a a
1
a b f 1 − t a b tb dt
−
1|
tf 1 − ta t
|t
2
2
0
⎡
1−1/q 1 1
1/q
b − a⎣
a b q
q/q−1
≤
t
dt
f 1 − ta t 2
dt
4
≤
b−a
4
0
1
1 − t
0
0
q/q−1
⎤
1−1/q 1 q 1/q
a
b
f 1 − t
⎦
tb dt
dt
2
3.20
0
⎡
1/q
1
q2−t/2s qt/2s
b − a q − 1 1−1/q ⎣
f a
f b
≤
dt
4
2q − 1
0
1
1/q ⎤
⎦.
q1−t/2s q1t/2s
f a
f b
dt
0
i If |f a| ≤ 1, we have
1
1
0
0
sq sq sq q2−t/2s qt/2s
f a
f b
,
,
dt ≤ f a g3 α
2 2
q1−t/2s q1t/2s
sq/2 sq sq f a
f b
,
,
dt ≤ f af b
g3 α
2 2
3.21
Abstract and Applied Analysis
11
ii If |f b| ≤ 1 ≤ |f a|, we have
1
q2−t/2s qt/2s
q/s
q sq
f a
f b
,
,
dt ≤ f a g3 α
2s 2
0
1
q/2s sq/2
q1−t/2s q1t/2s
q sq
f a
f b
f b
,
.
dt ≤ f a
g3 α
2s 2
0
3.22
iii If 1 ≤ |f b|, we have
1
q/s
q2−t/2s qt/2s
q q
f a
f b
,
,
dt ≤ f a
g3 α
2s 2s
0
1
q/2s
q1−t/2s q1t/2s
q q
f a
f b
,
.
dt ≤ f af b
g3 α
2s 2s
0
3.23
From 3.20 to 3.23, 3.17 holds.
2 Since |f x|q is s-geometrically convex and monotonically decreasing on a, b,
from Lemma 2.1 and Hölder inequality, we have
b
fa fb
1
−
fxdx
2
b−a a
⎡
1/q
1
q2−t/2s qt/2s
b − a q − 1 1−1/q ⎣
f a
f b
≤
dt
4
2q − 1
0
1
3.24
1/q ⎤
⎦.
q1−t/2s q1t/2s
f a
f b
dt
0
From 3.24 and 3.21 to 3.23, 3.18 holds. This completes the proof.
If taking s 1 in Theorem 3.3, we can derive the following corollary.
Corollary 3.4. Letf : I ⊂ R → R be differentiable on I ◦ , a, b ∈ I, with a < b, and f ∈
La, b. If |f x|q is geometrically convex and monotonically decreasing on a, b for q > 1, then
b
a b
b − a q − 1 1−1/q 1
fxdx ≤
G2 1, q; g3 α ,
−
f
2
b−a a
4
2q − 1
b
b − a q − 1 1−1/q fa fb
1
−
fxdx ≤
G2 1, q; g3 α ,
2
b−a a
4
2q − 1
where αu, v, G2 s, q; g3 α, and g3 α are the same as in Theorem 3.3.
3.25
12
Abstract and Applied Analysis
4. Application to Special Means
Let
ab
b−a
,
La, b b,
a /
2
ln b − ln a
1/p
bp1 − ap1
, a/
b, p ∈ R, p /
0, −1
Lp a, b p 1 b − a
Aa, b 4.1
be the arithmetic, logarithmic, generalized logarithmic means for a, b > 0 respectively.
Let fx xs /s, x ∈ 0, 1, 0 < s < 1, q ≥ 1, and then the function
q
f x xs−1q
4.2
is monotonically decreasing on 0, 1. For λ ∈ 0, 1, we have
s − 1q 1 − λs − 1 − λ ≤ 0.
s − 1qλs − λ ≤ 0,
4.3
Hence, |f x|q is s-geometrically convex on 0, 1 for 0 < s < 1.
Theorem 4.1. Let 0 < a < b ≤ 1, 0 < s < 1, and q ≥ 1. Then
1/q
1−1/q 4s
s
Aa, bs − Ls a, bs ≤ b − a
La, b
8
1 − sq
s−1q/2s−1
× as−1/2s Ls−1q/2s−1 a, b
La, b − bs−1q/2s
b
s−1/2s
s−1q/2s−1
as−1q/2s − Ls−1q/2s−1 a, b
La, b
1/q
1/q
Aas , bs − Ls a, bs 4.4
1/q
4s
b − a1−1/q s
≤
La, b
8
1 − sq
s−1q/2s−1
× bs−1/2s Ls−1q/2s−1 a, b
La, b − bs−1q/2s
s−1/2s
a
s−1q/2s
a
− Ls−1q/2s−1 a, b
s−1q/2s−1
1/q
La, b
1/q
.
Abstract and Applied Analysis
13
In particular, if q 1, one has
Aa, bs − Ls a, bs ≤ b − as Ls−1/2s−1 a, b s−1/s−2 La, b2
4
Aas , bs − Ls a, bs ≤
4.5
s−1/s−1 s−1/s−2
b − as
La, b 2 Ls−1/s−1 a, b
− Ls−1/2s−1 a, b
La, b .
4
Proof. Let fx xs /s, x ∈ 0, 1, 0 < s < 1. Then |f a| as−1 > bs−1 |f b| ≥ 1 and
1/q
1/s
q q
f a
,
g1 α
2s 2s
1/q 1/q
s−1q/2s−1
−1/q s−1/2s 2sLa, b
b − a
a
La, b − bs−1q/2s
,
Ls−1q/2s−1 a, b
1 − sq
1/q
f af b1/2s g2 α q , q
2s 2s
1/q
s−1q/2s−1
2sLa, b 1/q s−1q/2s a
b −a−1/q bs−1/2s
− Ls−1q/2s−1 a, b
La, b
,
1 − sq
1/q
f af b1/2s g1 α q , q
2s 2s
1/q
s−1q/2s−1
2sLa, b 1/q
b − a−1/q bs−1/2s
La, b− bs−1q/2s
,
Ls−1q/2s−1 a, b
1 − sq
1/q
1/s
q q
f a
,
g2 α
2s 2s
1/q
s−1q/2s−1
2sLa, b 1/q s−1q/2s a
b − a−1/q as−1/2s
− Ls−1q/2s−1 a, b
La, b
.
1 − sq
4.6
By Theorem 3.1, Theorem 4.1 is thus proved.
Theorem 4.2. Let 0 < a < b ≤ 1, s ∈ 0, 1, and q > 1. Then one has
1−1/q Aa, bs − Ls a, bs ≤ b − as q − 1
A as−1/2s , bs−1/2s
2
2q − 1
1/q
s−1q/2s−1
× Ls−1q/2s−1 a, b
La, b
14
Abstract and Applied Analysis
1−1/q Aas , bs − Ls a, bs ≤ b − as q − 1
A as−1/2s , bs−1/2s
2
2q − 1
1/q
s−1q/2s−1
× Ls−1q/2s−1 a, b
La, b
.
4.7
Proof. Let fx xs /s, x ∈ 0, 1, 0 < s < 1. Then |f a| as−1 > bs−1 |f b| ≥ 1 and
s−1q/2s−1
q q
g3 α
,
a−s−1q/2s Ls−1q/2s−1 a, b
La, b.
2s 2s
4.8
Using Theorem 3.3, Theorem 4.2 is thus proved.
Acknowledgments
The research was supported by Science Research Funding of Inner Mongolia University for
Nationalities Project no. NMD1103.
References
1 H. Hudzik and L. Maligranda, “Some remarks on s-convex functions,” Aequationes Mathematicae, vol.
48, no. 1, pp. 100–111, 1994.
2 S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to
special means of real numbers and to trapezoidal formula,” Applied Mathematics Letters, vol. 11, no. 5,
pp. 91–95, 1998.
3 U. S. Kirmaci, “Inequalities for differentiable mappings and applications to special means of real
numbers and to midpoint formula,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 137–146,
2004.
4 U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, and J. Pečarić, “Hadamard-type inequalities for sconvex functions,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 26–35, 2007.
5 S. Hussain, M. I. Bhatti, and M. Iqbal, “Hadamard-type inequalities for s-convex functions. I,” Punjab
University. Journal of Mathematics, vol. 41, pp. 51–60, 2009.
6 M. W. Alomari, M. Darus, and U. S. Kirmaci, “Some inequalities of Hermite-Hadamard type for sconvex functions,” Acta Mathematica Scientia B, vol. 31, no. 4, pp. 1643–1652, 2011.
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