# New Hermite-Hadamard Type Inequalities forn-Times Differentiable and s-Logarithmically Preinvex Functions

Authors: Shuhong Wang; Ximin Liu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The concept ofs-logarithmically preinvex function is introduced, and by creating an integral identity involving an n-times differentiable function, some new Hermite-Hadamard type inequalities for s-logarithmically preinvex functions are established.


---

## Body

## 1. Introduction

Throughout this paper, letR
=
(
-
∞
,
∞
), R
+
=
(
0
,
∞
), N denote the set of all positive integers, I denote the interval in R, and A denote the set in R
n
,
n
∈
N.Let us recall some definitions of various convex functions.Definition 1.
A functionf
:
I
⊆
R
→
R is said to be convex if
(1)
f
(
λ
x
+
(
1
-
λ
)
y
)
≤
λ
f
(
x
)
+
(
1
-
λ
)
f
(
y
)
holds for all x
,
y
∈
I and λ
∈
[
0,1
]. If inequality (1) reverses, then f is said to be concave on I.Definition 2 (see [1]).
A setA
⊆
R
n is said to be invex with respect to the map η
:
A
×
A
→
R
n, if for every x
,
y
∈
A and t
∈
[
0,1
]
(2)
y
+
t
η
(
x
,
y
)
∈
A
.
The invex set A is also called a η-connected set.It is obvious that every convex set is invex with respect to the mapη
(
x
,
y
)
=
x
-
y, but there exist invex sets which are not convex (see [1], e.g.).Definition 3 (see [1]).
LetA
⊆
R
n be an invex set with respect to η
:
A
×
A
→
R
n. For every x
,
y
∈
A, the η-path P
x
v joining the points x and v
=
x
+
η
(
y
,
x
) is defined by
(3)
P
x
v
=
{
z
∣
z
=
x
+
t
η
(
y
,
x
)
,
t
∈
[
0,1
]
}
.Definition 4 (see [2]).
LetA
⊆
R
n be an invex set with respect to η
:
A
×
A
→
R
n. A function f
:
A
→
R is said to be preinvex with respect to η, if for every x
,
y
∈
A and t
∈
[
0,1
]
(4)
f
(
y
+
t
η
(
x
,
y
)
)
≤
t
f
(
x
)
+
(
1
-
t
)
f
(
y
)
.
The function f is said to be preincave if and only if -
f is preinvex.Every convex function is preinvex with respect to the mapη
(
x
,
y
)
=
x
-
y, but not conversely (see [2], e.g.).Definition 5 (see [3]).
LetA
⊆
R
n be an invex set with respect to η
:
A
×
A
→
R
n. The function f
:
A
→
R
+ on the set A is said to be logarithmically preinvex with respect to η, if for every x
,
y
∈
A and t
∈
[
0,1
]
(5)
f
(
y
+
t
η
(
x
,
y
)
)
≤
[
f
(
x
)
]
t
[
f
(
y
)
]
1
-
t
.For properties and applications of preinvex and logarithmically preinvex functions, please refer to [1–8] and closely related references therein.The most important inequality in the theory of convex functions, the well known Hermite-Hadamard’s integral inequality, may be stated as follows. LetI
⊆
R and a
,
b
∈
I with a
<
b. If f
:
[
a
,
b
]
⊆
I
→
R is a convex function on [
a
,
b
], then
(6)
f
(
a
+
b
2
)
≤
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
≤
f
(
a
)
+
f
(
b
)
2
.
If f is concave on [
a
,
b
], then inequality (6) is reversed.Inequality (6) has been generalized by many mathematicians. Some of them may be recited as follows.Theorem 6 (see [9, Theorem 2.2]).
Letf
:
I
⊆
R
→
R be a differentiable mapping on I and a
,
b
∈
I with a
<
b. If |
f
′
(
x
)
| is convex on [
a
,
b
], then
(7)
|
f
(
a
)
+
f
(
b
)
2
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
(
b
-
a
)
[
|
f
′
(
a
)
|
+
|
f
′
(
b
)
|
]
8
.Theorem 7 (see [10, Theorem 1]).
LetI
⊆
R and a
,
b
∈
I with a
<
b. If f
:
[
a
,
b
]
⊆
I
→
R is differentiable on [
a
,
b
] such that |
f
′
(
x
)
|
q is a convex function on [
a
,
b
] for q
≥
1, then
(8)
|
f
(
a
)
+
f
(
b
)
2
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
4
[
|
f
′
(
a
)
|
q
+
|
f
′
(
b
)
|
q
2
]
1
/
q
.Theorem 8 (see [11, Theorem 2.3]).
Letf
:
I
⊆
R
→
R be differentiable on I, a
,
b
∈
I with a
<
b and p
>
1. If |
f
′
(
x
)
|
p
/
(
p
-
1
) is convex on [
a
,
b
], then
(9)
|
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
16
(
4
p
+
1
)
1
/
p
×
{
[
|
f
′
(
a
)
|
p
/
(
p
-
1
)
+
3
|
f
′
(
b
)
|
p
/
(
p
-
1
)
]
1
-
1
/
p
mm
+
[
3
|
f
′
(
a
)
|
p
/
(
p
-
1
)
+
|
f
′
(
b
)
|
p
/
(
p
-
1
)
]
1
-
1
/
p
}
.Theorem 9 (see [6]).
LetA
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. If f
:
A
→
R
+ is a preinvex function on A, then the following inequality holds:
(10)
f
(
2
b
+
η
(
a
,
b
)
2
)
≤
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
≤
f
(
a
)
+
f
(
b
)
2
.Theorem 10 (see [4, Theorem 4.3]).
LetA
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is a twice differentiable function on A and |
f
′′
(
x
)
| is preinvex on A. If q
>
1 and f
′′ is integrable on the η-path  P
b
c  for c
=
b
+
η
(
a
,
b
), then
(11)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
|
≤
[
η
(
a
,
b
)
]
2
12
(
1
2
)
1
/
q
[
|
f
′′
(
a
)
|
q
+
|
f
′′
(
b
)
|
q
]
1
/
q
.Theorem 11 (see [12, Theorem 3.1]).
Forn
∈
N and n
≥
2, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is an n-times differentiable function on A and f
(
n
) is integrable on the η-path P
b
c for c
=
b
+
η
(
a
,
b
). If |
f
(
n
)
|
q is preinvex on A for q
≥
1, then
(12)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
1
n
-
1
‍
[
η
(
a
,
b
)
]
k
(
1
-
k
)
4
[
(
k
+
1
)
!
]
m
m
m
×
[
f
(
k
)
(
b
)
+
(
-
1
)
k
f
(
k
)
(
b
+
η
(
a
,
b
)
)
]
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
|
η
(
a
,
b
)
|
n
(
n
-
1
)
1
-
1
/
q
4
[
(
n
+
1
)
!
]
(
n
+
2
)
1
/
q
×
{
[
n
|
f
(
n
)
(
a
)
|
q
+
(
n
2
-
2
)
|
f
(
n
)
(
b
)
|
q
]
1
/
q
mm
+
[
(
n
2
-
2
)
|
f
(
n
)
(
a
)
|
q
+
n
|
f
(
n
)
(
b
)
|
q
]
1
/
q
}
.Theorem 12 (see [13, Theorem 5]).
Forn
∈
N and n
≥
2, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is a function such that f
(
n
) exits on A and f
(
n
) is integrable on [
a
,
a
+
η
(
b
,
a
)
]. If |
f
(
n
)
|
q is logarithmically preinvex on  A  for q
≥
1, then we have the inequality
(13)
|
∑
k
=
2
n
-
1
‍
(
-
1
)
k
(
k
-
1
)
(
η
(
b
,
a
)
)
k
2
(
k
+
1
)
!
f
(
a
)
+
f
(
a
+
η
(
b
,
a
)
)
2
-
1
η
(
b
,
a
)
∫
a
a
+
η
(
b
,
a
)
‍
f
(
x
)
d
x
-
∑
k
=
2
n
-
1
‍
(
-
1
)
k
(
k
-
1
)
(
η
(
b
,
a
)
)
k
2
(
k
+
1
)
!
f
(
k
)
(
a
+
η
(
b
,
a
)
)
|
≤
(
η
(
b
,
a
)
)
n
2
n
!
(
n
-
1
n
+
1
)
1
-
1
/
q
[
E
1
(
n
,
q
)
]
1
/
q
,
where
(14)
E
1
(
n
,
q
)
=
(
-
1
)
n
n
!
{
q
[
ln
⁡
(
|
f
(
n
)
(
b
)
|
)
-
ln
⁡
(
|
f
(
n
)
(
a
)
|
)
]
+
2
}
|
f
(
n
)
(
a
)
|
q
q
n
+
1
[
ln
⁡
(
|
f
(
n
)
(
b
)
|
)
-
ln
⁡
(
|
f
(
n
)
(
a
)
|
)
]
n
+
1
-
2
|
f
(
n
)
(
b
)
|
q
q
[
ln
⁡
(
|
f
(
n
)
(
b
)
|
)
-
ln
⁡
(
|
f
(
n
)
(
a
)
|
)
]
-
n
!
|
f
(
n
)
(
b
)
|
q
×
∑
k
=
1
n
‍
(
-
1
)
k
{
q
[
ln
⁡
(
|
f
(
n
)
(
b
)
|
)
-
ln
⁡
(
|
f
(
n
)
(
a
)
|
)
]
+
2
}
(
n
-
k
)
!
q
k
+
1
[
ln
⁡
(
|
f
(
n
)
(
b
)
|
)
-
ln
⁡
(
|
f
(
n
)
(
a
)
|
)
]
k
+
1
.Recently, some related inequalities for preinvex functions were also obtained in [14, 15].In the paper, the concept ofs-logarithmically preinvex function is introduced, and by creating an integral identity involving an n-times differentiable function, some new Hermite-Hadamard type inequalities for s-logarithmically preinvex functions are established which generalize some known results.


## 2.  New Definition and Lemmas

Now we introduce concepts ofs-logarithmically preinvex functions.Definition 13.
LetA
⊆
R
n be an invex set with respect to η
:
A
×
A
→
R
n. The function f
:
A
→
R
+ on the set A is said to be s-logarithmically preinvex with respect to η, if for every x
,
y
∈
A
,
t
∈
[
0,1
], and some s
∈
(
0,1
]
(15)
f
(
y
+
t
η
(
x
,
y
)
)
≤
[
f
(
x
)
]
t
s
[
f
(
y
)
]
(
1
-
t
)
s
.
Clearly, when takings
=
1 in (15), then f becomes the standard logarithmically convex function on A.
In order to obtain our main results, we need the following lemmas.Lemma 14.
Forn
∈
N, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0  for all a
≠
b. If f
:
A
→
R is an n-times differentiable function on A and f
(
n
) is integrable on the η-path P
b
c  for c
=
b
+
η
(
a
,
b
), then
(16)
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
=
[
η
(
a
,
b
)
]
n
2
n
!
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
-
t
)
n
-
1
(
2
t
-
n
)
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
mm
+
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
m
m
m
m
m
m
×
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
,
where x
∈
[
b
,
b
+
η
(
b
,
a
)
] and the above summation is zero for n
=
1.Proof.
Sincea
,
b
∈
A and A is an invex set with respect to η, for every t
∈
[
0,1
], we have b
+
t
η
(
a
,
b
)
∈
A. When n
=
1, by integrating by part in the right-hand side of (16), one gives
(17)
η
(
a
,
b
)
2
[
∫
0
1
‍
(
2
t
-
1
)
f
′
(
b
+
t
η
(
a
,
b
)
)
d
t
]
=
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
.
Hence, the identity (16) holds for n
=
1.
Whenn
=
m and m
≥
1, suppose that the identity (16) is valid.
Whenn
=
m
+
1, by the hypothesis, we have
(18)
[
η
(
a
,
b
)
]
m
+
1
2
(
m
+
1
)
!
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
-
t
)
m
(
2
t
-
m
-
1
)
m
m
×
f
(
m
+
1
)
(
b
+
t
η
(
a
,
b
)
)
d
t
mm
+
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
m
(
2
t
+
m
-
1
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
f
(
m
+
1
)
(
b
+
t
η
(
a
,
b
)
)
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
=
[
η
(
a
,
b
)
]
m
2
(
m
+
1
)
!
×
[
(
b
-
x
η
(
a
,
b
)
)
m
(
2
x
-
2
b
η
(
a
,
b
)
-
m
-
1
)
f
(
m
)
(
x
)
mm
m
-
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
-
t
)
m
-
1
(
m
+
1
)
(
m
-
2
t
)
m
m
m
m
×
f
(
m
)
(
b
+
t
η
(
a
,
b
)
)
d
t
mm
m
-
(
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
m
(
2
x
-
2
b
η
(
a
,
b
)
+
m
-
1
)
f
(
m
)
(
x
)
mm
m
-
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
m
-
1
(
m
+
1
)
m
m
m
m
×
(
2
-
2
t
-
m
)
f
(
m
)
(
b
+
t
η
(
a
,
b
)
)
d
t
(
b
-
x
η
(
a
,
b
)
)
m
]
=
1
2
(
m
+
1
)
!
η
(
a
,
b
)
×
{
(
b
+
η
(
a
,
b
)
-
x
)
m
(
b
-
x
)
m
[
2
x
-
2
b
-
(
m
+
1
)
η
(
a
,
b
)
]
mm
-
(
b
+
η
(
a
,
b
)
-
x
)
m
[
2
x
-
2
b
+
(
m
-
1
)
η
(
a
,
b
)
]
}
×
f
(
m
)
(
x
)
+
[
η
(
a
,
b
)
]
m
2
m
!
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
-
t
)
m
-
1
(
2
t
-
m
)
f
(
m
)
(
b
+
t
η
(
a
,
b
)
)
d
t
mm
+
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
m
-
1
(
2
t
+
m
-
2
)
m
m
m
m
m
×
f
(
m
)
(
b
+
t
η
(
a
,
b
)
)
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
=
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
m
+
1
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
m
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
]
×
f
(
k
-
1
)
(
x
)
.
Therefore, whenn
=
m
+
1, the identity (16) holds. By induction, the proof of Lemma 14 is complete.Remark 15.
Under the conditions of Lemma14, we have
(19)
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
[
η
(
a
,
b
)
]
k
-
1
(
2
-
k
)
2
k
!
f
(
k
-
1
)
(
b
)
=
[
η
(
a
,
b
)
]
n
2
n
!
∫
0
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
m
m
m
m
m
m
m
m
m
×
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
,
(20)
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
[
η
(
a
,
b
)
]
k
-
1
(
1
-
k
)
[
1
+
(
-
1
)
k
-
1
]
2
k
k
!
×
f
(
k
-
1
)
(
b
+
η
(
a
,
b
)
2
)
=
[
η
(
a
,
b
)
]
n
2
n
!
×
[
∫
0
1
/
2
‍
(
-
t
)
n
-
1
(
2
t
-
n
)
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
m
m
+
∫
1
/
2
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
∫
0
1
/
2
‍
]
,
(21)
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
[
-
η
(
a
,
b
)
]
k
-
1
(
2
-
k
)
2
k
!
f
(
k
-
1
)
(
b
+
η
(
a
,
b
)
)
=
[
η
(
a
,
b
)
]
n
2
n
!
∫
0
1
‍
(
-
t
)
n
-
1
(
2
t
-
n
)
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
d
t
.Proof.
These are special cases of Lemma14 for x
=
b,  b
+
η
(
a
,
b
)
/
2, b
+
η
(
a
,
b
), respectively.Remark 16.
Adding the identities (19) and (21) and then dividing by 2 result in Lemma 14 from [12].Lemma 17.
Letμ
>
0 and x
≥
0. Then
(22)
E
(
n
;
μ
,
x
)
≜
∫
0
x
‍
t
n
μ
t
d
t
=
{
(
-
1
)
n
+
1
n
!
(
ln
⁡
μ
)
n
+
1
+
n
!
μ
x
∑
k
=
0
n
‍
(
-
1
)
k
x
n
-
k
(
n
-
k
)
!
(
ln
⁡
μ
)
k
+
1
,
μ
≠
1
,
x
n
+
1
n
+
1
,
μ
=
1
for n
≥
0, n
∈
N.Proof.
Whenμ
=
1, the proof is straightforward.
Whenμ
≠
1, for n
=
0, we have
(23)
∫
0
x
‍
μ
t
d
t
=
μ
x
-
1
ln
⁡
μ
,
which coincides with the right-hand side of (22) for n
=
0.
Forn
=
1, we get
(24)
∫
0
x
‍
t
μ
t
d
t
=
x
μ
x
ln
⁡
μ
-
1
ln
⁡
μ
∫
0
x
‍
μ
t
d
t
=
1
(
ln
⁡
μ
)
2
+
μ
x
[
x
ln
⁡
μ
-
1
(
ln
⁡
μ
)
2
]
,
which coincides with the right-hand side of (22) for n
=
1.
Suppose that (22) is true for n
=
m,  m
≥
0, then, for n
=
m
+
1, it follows that
(25)
∫
0
x
‍
t
m
+
1
μ
t
d
t
=
x
m
+
1
μ
x
ln
⁡
μ
-
m
+
1
ln
⁡
μ
∫
0
x
‍
t
m
μ
t
d
t
=
x
m
+
1
μ
x
ln
⁡
μ
-
m
+
1
ln
⁡
μ
[
(
-
1
)
m
+
1
m
!
(
ln
⁡
μ
)
m
+
1
mmmmmmmmmmm
+
m
!
μ
x
∑
k
=
0
m
‍
(
-
1
)
k
x
m
-
k
(
m
-
k
)
!
(
ln
⁡
μ
)
k
+
1
]
=
(
-
1
)
m
+
2
(
m
+
1
)
!
(
ln
⁡
μ
)
m
+
2
+
x
m
+
1
μ
x
ln
⁡
μ
+
(
m
+
1
)
!
μ
x
∑
k
=
1
m
+
1
‍
(
-
1
)
k
x
m
-
k
+
1
(
m
-
k
+
1
)
!
(
ln
⁡
μ
)
k
+
1
=
(
-
1
)
m
+
2
(
m
+
1
)
!
(
ln
⁡
μ
)
m
+
2
+
(
m
+
1
)
!
μ
x
∑
k
=
0
m
+
1
‍
(
-
1
)
k
x
m
-
k
+
1
(
m
-
k
+
1
)
!
(
ln
⁡
μ
)
k
+
1
.
Therefore, when n
=
m
+
1, the identity (22) holds. By induction, the proof of Lemma 17 is complete.Lemma 18 (see [16]).
Letγ
>
0, μ
>
0, and x
≥
0. Then
(26)
G
(
γ
;
μ
,
x
)
≜
∫
0
x
‍
t
γ
-
1
μ
t
d
t
=
x
γ
μ
x
∑
k
=
1
∞
‍
(
-
x
ln
⁡
μ
)
k
-
1
(
γ
)
k
<
∞
,
where (
γ
)
k
=
γ
(
γ
+
1
)
(
γ
+
2
)
⋯
(
γ
+
k
-
1
).By Lemmas17 and 18, a straightforward computation gives the following lemmas.Lemma 19.
Letμ
>
0 and x
≥
0. Then
(27)
F
(
n
;
μ
,
x
)
≜
n
E
(
n
-
1
;
μ
,
x
)
-
2
E
(
n
;
μ
,
x
)
=
{
2
μ
x
x
n
ln
⁡
μ
+
n
!
(
ln
⁡
μ
+
2
)
×
[
(
-
1
)
n
(
ln
⁡
μ
)
n
+
1
+
μ
x
∑
k
=
1
n
‍
(
-
1
)
k
-
1
x
n
-
k
(
n
-
k
)
!
(
ln
⁡
μ
)
k
+
1
]
,
μ
≠
1
,
x
n
(
n
+
1
-
2
x
)
n
+
1
,
μ
=
1
for n
≥
0, n
∈
N.Lemma 20.
Letγ
>
0, μ
>
0, and x
≥
0. Then
(28)
H
(
γ
;
μ
,
x
)
≜
n
G
(
γ
;
μ
,
x
)
-
2
G
(
γ
+
1
;
μ
,
x
)
=
x
γ
μ
x
∑
k
=
1
∞
‍
(
-
x
ln
⁡
μ
)
k
-
1
[
n
(
γ
+
k
)
-
2
x
γ
]
(
γ
)
k
+
1
,
where (
γ
)
k
+
1
=
γ
(
γ
+
1
)
(
γ
+
2
)
⋯
(
γ
+
k
).Lemma 21 (see [17]).
Let0
<
ϕ
≤
1
≤
ψ and 0
<
t, s
≤
1. Then
(29)
ϕ
t
s
≤
ϕ
s
t
,
ψ
t
s
≤
ψ
s
t
+
1
-
s
.


## 3. Hermite-Hadamard Type Inequalities

Now we start out to establish some new Hermite-Hadamard type inequalities forn-times differentiable and s-logarithmically preinvex functions.Theorem 22.
Forn
∈
N and n
≥
2, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is an n-times differentiable function on A and f
(
n
) is integrable on the η-path P
b
c for c
=
b
+
η
(
a
,
b
). If  |
f
(
n
)
|
q is s-logarithmically preinvex on A for q
≥
1, then for x
∈
[
b
,
b
+
η
(
a
,
b
)
] and some s
∈
(
0,1
]
(30)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
m
m
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
b
-
x
)
k
-
1
]
×
f
(
k
-
1
)
(
x
)
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
|
≤
[
η
(
a
,
b
)
]
(
n
+
1
-
q
)
/
q
2
n
!
(
n
+
1
)
1
-
1
/
q
×
{
[
F
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
m
m
×
[
(
x
-
b
)
n
[
(
n
+
1
)
η
(
a
,
b
)
-
2
x
+
2
b
]
]
1
-
1
/
q
m
m
×
[
F
(
n
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
δ
|
f
(
n
)
(
b
)
|
θ
m
m
×
[
(
b
+
η
(
a
,
b
)
-
x
)
n
[
(
n
-
1
)
η
(
a
,
b
)
+
2
x
-
2
b
]
]
1
-
1
/
q
mm
×
[
F
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
,
where ξ
=
|
f
(
n
)
(
a
)
/
f
(
n
)
(
b
)
|, F
(
n
;
μ
,
x
) is defined in Lemma 19, and
(31)
{
α
,
β
}
=
{
{
0
,
s
}
,
if
0
<
|
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
≤
1
,
{
0,1
}
,
if
0
<
|
f
(
n
)
(
a
)
|
<
1
<
|
f
(
n
)
(
b
)
|
,
{
1
-
s
,
s
}
,
if
0
<
|
f
(
n
)
(
b
)
|
<
1
<
|
f
(
n
)
(
a
)
|
,
{
1
-
s
,
1
}
,
if
|
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
>
1
,
{
δ
,
θ
}
=
{
{
s
,
0
}
,
if
0
<
|
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
≤
1
,
{
s
,
1
-
s
}
,
if
0
<
|
f
(
n
)
(
a
)
|
<
1
<
|
f
(
n
)
(
b
)
|
,
{
1,0
}
,
if
0
<
|
f
(
n
)
(
b
)
|
<
1
<
|
f
(
n
)
(
a
)
|
,
{
1,1
-
s
}
,
if
|
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
>
1
.Proof.
Sincea
,
b
∈
A and A is an invex set with respect to η, for every t
∈
[
0,1
], we have b
+
t
η
(
a
,
b
)
∈
A. Using Lemma 14, Hölder’s inequality, and s-logarithmically preinvexity of |
f
(
n
)
|
q, it yields that
(32)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
m
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
]
×
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
2
n
!
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
|
d
t
mm
+
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
m
m
m
m
m
m
×
|
f
(
n
)
(
b
+
t
η
(
a
,
b
)
)
|
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
≤
[
η
(
a
,
b
)
]
n
2
n
!
×
{
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
d
t
]
1
-
1
/
q
m
m
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
m
m
m
m
×
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
1
/
q
m
m
+
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
d
t
]
1
-
1
/
q
m
m
×
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
m
m
m
m
m
×
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
]
1
/
q
}
.
By Lemmas 17 and 21, for 0
<
|
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
≤
1, we give
(33)
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
s
q
(
1
-
t
)
|
f
(
n
)
(
a
)
|
s
q
t
d
t
=
|
f
(
n
)
(
b
)
|
s
q
[
n
E
(
n
-
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
m
m
m
m
m
m
m
-
2
E
(
n
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
,
(34)
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
m
m
m
×
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
s
q
(
1
-
t
)
m
m
m
×
|
f
(
n
)
(
a
)
|
s
q
t
d
t
=
∫
0
(
b
+
η
(
a
,
b
)
-
x
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
s
q
t
|
f
(
n
)
(
a
)
|
s
q
(
1
-
t
)
d
t
=
|
f
(
n
)
(
a
)
|
s
q
[
n
E
(
n
-
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
mmmmmmm
-
2
E
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
.
For 0
<
|
f
(
n
)
(
a
)
|
<
1
<
|
f
(
n
)
(
b
)
|, we get
(35)
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
|
f
(
n
)
(
a
)
|
s
q
t
d
t
=
|
f
(
n
)
(
b
)
|
q
[
n
E
(
n
-
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
m
m
m
m
m
m
-
2
E
(
n
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
,
(36)
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
m
m
m
×
|
f
(
n
)
(
a
)
|
s
q
t
d
t
=
∫
0
(
b
+
η
(
a
,
b
)
-
x
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
s
t
+
1
-
s
)
m
m
m
×
|
f
(
n
)
(
a
)
|
s
q
(
1
-
t
)
d
t
=
|
f
(
n
)
(
a
)
|
s
q
|
f
(
n
)
(
b
)
|
q
(
1
-
s
)
[
n
E
(
n
-
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
mmmmmmmmmmmm
-
2
E
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
.
For 0
<
|
f
(
n
)
(
b
)
|
<
1
<
|
f
(
n
)
(
a
)
|, we obtain
(37)
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
s
(
1
-
t
)
|
f
(
n
)
(
a
)
|
q
(
s
t
+
1
-
s
)
d
t
=
|
f
(
n
)
(
b
)
|
q
s
|
f
(
n
)
(
a
)
|
q
(
1
-
s
)
[
n
E
(
n
-
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
-
2
E
(
n
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
,
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
s
(
1
-
t
)
m
m
m
×
|
f
(
n
)
(
a
)
|
q
(
s
t
+
1
-
s
)
d
t
=
∫
0
(
b
+
η
(
a
,
b
)
-
x
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
s
t
m
m
m
×
|
f
(
n
)
(
a
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
d
t
=
|
f
(
n
)
(
a
)
|
q
[
n
E
(
n
-
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
mmmmmm
-
2
E
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
.
For |
f
(
n
)
(
a
)
|
,
|
f
(
n
)
(
b
)
|
>
1, we have
(38)
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
m
m
m
×
|
f
(
n
)
(
a
)
|
q
(
s
t
+
1
-
s
)
d
t
=
|
f
(
n
)
(
b
)
|
q
|
f
(
n
)
(
a
)
|
q
(
1
-
s
)
[
n
E
(
n
-
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
-
2
E
(
n
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
,
(39)
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
≤
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
n
-
1
(
2
t
+
n
-
2
)
×
|
f
(
n
)
(
b
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
|
f
(
n
)
(
a
)
|
q
(
s
t
+
1
-
s
)
d
t
=
∫
0
(
b
+
η
(
a
,
b
)
-
x
)
/
(
η
(
a
,
b
)
)
‍
t
n
-
1
(
n
-
2
t
)
×
|
f
(
n
)
(
b
)
|
q
(
s
t
+
1
-
s
)
|
f
(
n
)
(
a
)
|
q
[
s
(
1
-
t
)
+
1
-
s
]
d
t
=
|
f
(
n
)
(
a
)
|
q
|
f
(
n
)
(
b
)
|
q
(
1
-
s
)
[
n
E
(
n
-
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
mmmmmmmlmmmm
h
-
2
E
(
n
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
.
Using Lemma 19 and substituting (33) to (39) into (32), we get inequality (30).
Theorem22 is thus proved.Corollary 23.
Under the assumptions of Theorem22,
(1) ifq
=
1, then
(40)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mh
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
2
n
!
{
[
F
(
n
;
ξ
-
s
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
[
F
(
n
;
ξ
s
,
x
-
b
η
(
a
,
b
)
)
]
mmmmmm
+
|
f
(
n
)
(
a
)
|
δ
|
f
(
n
)
(
b
)
|
θ
m
m
m
m
m
m
×
[
F
(
n
;
ξ
-
s
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
}
;
(2) ifs
=
1, then
(41)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
(
n
+
1
-
q
)
/
q
2
n
!
(
n
+
1
)
1
-
1
/
q
×
{
[
F
(
n
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
b
)
|
[
(
x
-
b
)
n
[
(
n
+
1
)
η
(
a
,
b
)
-
2
x
+
2
b
]
]
1
-
1
/
q
m
m
×
[
F
(
n
;
ξ
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
m
m
×
[
(
b
+
η
(
a
,
b
)
-
x
)
n
[
(
n
-
1
)
η
(
a
,
b
)
+
2
x
-
2
b
]
]
1
-
1
/
q
m
m
×
[
F
(
n
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
.Theorem 24.
Forn
∈
N and n
≥
2, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is an n-times differentiable function on A and f
(
n
) is integrable on the η-path P
b
c for c
=
b
+
η
(
a
,
b
). If |
f
(
n
)
|
q is s-logarithmically preinvex on A for q
>
1, then for x
∈
[
b
,
b
+
η
(
a
,
b
)
] and some s
∈
(
0,1
]
(42)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
-
2
+
1
/
q
2
2
-
1
/
q
n
!
(
1
-
q
2
q
-
1
)
1
-
1
/
q
×
{
[
G
(
n
q
-
q
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
[
(
n
η
(
a
,
b
)
)
2
q
-
1
/
q
-
1
(
n
η
(
a
,
b
)
-
2
x
+
2
b
)
(
2
q
-
1
)
/
(
q
-
1
)
m
m
m
-
(
n
η
(
a
,
b
)
)
(
2
q
-
1
)
/
(
q
-
1
)
]
1
-
1
/
q
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
mm
×
[
G
(
n
q
-
q
+
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
mm
+
[
(
(
n
-
2
)
η
(
a
,
b
)
+
2
x
-
2
b
)
(
2
q
-
1
)
/
(
q
-
1
)
mmm
m
-
(
n
η
(
a
,
b
)
)
(
2
q
-
1
)
/
(
q
-
1
)
]
1
-
1
/
q
mm
×
|
f
(
n
)
(
a
)
|
δ
|
f
(
n
)
(
b
)
|
θ
m
m
×
[
G
(
n
q
-
q
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
,
where ξ, {
α
,
β
} and {
δ
,
θ
} are given in Theorem 22 and G
(
γ
;
μ
,
x
) is defined by (26).Proof.
Sincea
,
b
∈
A and A is an invex set with respect to η, for every t
∈
[
0,1
], we have b
+
t
η
(
a
,
b
)
∈
A. Using Lemma 14, Hölder’s inequality, and s-logarithmically preinvexity of |
f
(
n
)
|
q, it follows that
(43)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
2
n
!
×
{
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
n
-
2
t
)
q
/
(
q
-
1
)
d
t
]
1
-
1
/
q
mm
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
(
n
-
1
)
q
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
]
1
/
q
mm
+
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
2
t
+
n
-
2
)
q
/
(
q
-
1
)
d
t
]
1
-
1
/
q
mm
×
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
(
n
-
1
)
q
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
m
m
m
m
×
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
]
1
/
q
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
(
n
-
2
t
)
q
/
(
q
-
1
)
d
t
]
1
-
1
/
q
}
.
The rest is the same as the proof of Theorem22.Corollary 25.
Under the assumptions of Theorem24, if s
=
1, then
(44)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
-
2
+
1
/
q
2
2
-
1
/
q
n
!
(
1
-
q
2
q
-
1
)
1
-
1
/
q
×
{
[
G
(
n
q
-
q
+
1
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
[
(
n
η
(
a
,
b
)
-
2
x
+
2
b
)
(
2
q
-
1
)
/
(
q
-
1
)
mm
m
-
(
n
η
(
a
,
b
)
)
(
2
q
-
1
)
/
(
q
-
1
)
(
n
η
(
a
,
b
)
-
2
x
+
2
b
)
(
2
q
-
1
)
/
(
q
-
1
)
]
1
-
1
/
q
|
f
(
n
)
(
b
)
|
mm
×
[
G
(
n
q
-
q
+
1
;
ξ
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
mm
+
[
(
(
n
-
2
)
η
(
a
,
b
)
+
2
x
-
2
b
)
(
2
q
-
1
)
/
(
q
-
1
)
mmmm
-
(
n
η
(
a
,
b
)
)
(
2
q
-
1
)
/
(
q
-
1
)
]
1
-
1
/
q
|
f
(
n
)
(
a
)
|
mm
×
[
G
(
n
q
-
q
+
1
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
.Theorem 26.
Forn
∈
N and n
≥
2, let A
⊆
R be an open invex set with respect to η
:
A
×
A
→
R and a
,
b
∈
A with η
(
a
,
b
)
>
0 for all a
≠
b. Suppose that f
:
A
→
R is an n-times differentiable function on A and f
(
n
) is integrable on the η-path P
b
c for c
=
b
+
η
(
a
,
b
). If |
f
(
n
)
|
q is s-logarithmically preinvex on A for q
>
1, then for x
∈
[
b
,
b
+
η
(
a
,
b
)
], 0
≤
r
≤
(
n
-
1
)
q, and some s
∈
(
0,1
]
(45)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
(
r
+
2
-
q
)
/
q
2
n
!
[
q
-
1
(
n
q
-
r
-
1
)
(
n
q
+
q
-
r
-
2
)
]
1
-
1
/
q
×
{
[
H
(
r
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
(
x
-
b
)
(
n
q
-
r
-
1
)
/
q
mm
×
[
n
(
n
q
+
q
-
r
-
2
)
η
(
a
,
b
)
mm
mm
-
2
(
n
q
-
r
-
1
)
(
x
-
b
)
]
1
-
1
/
q
mm
×
[
H
(
r
+
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
δ
|
f
(
n
)
(
b
)
|
θ
mm
×
[
n
(
n
q
+
q
-
r
-
2
)
η
(
a
,
b
)
mm
m
m
-
2
(
n
q
-
r
-
1
)
(
b
+
η
(
a
,
b
)
-
x
)
]
1
-
1
/
q
mm
×
(
b
+
η
(
a
,
b
)
-
x
)
(
n
q
-
r
-
1
)
/
q
mm
×
[
H
(
r
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
,
where ξ, {
α
,
β
} and {
δ
,
θ
} are given in Theorem 22 and H
(
γ
;
μ
,
x
) is defined by (28).Proof.
Sincea
,
b
∈
A and A is an invex set with respect to η, for every t
∈
[
0,1
], we have b
+
t
η
(
a
,
b
)
∈
A. Using Lemma 14, Hölder’s inequality, and s-logarithmically preinvexity of |
f
(
n
)
|
q, it turns out that
(46)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
2
n
!
×
{
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
]
1
/
q
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
(
n
q
-
q
-
r
)
/
(
q
-
1
)
(
n
-
2
t
)
d
t
]
1
-
1
/
q
mm
×
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
r
(
n
-
2
t
)
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
m
m
m
m
×
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
]
1
/
q
mm
+
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
(
n
q
-
q
-
r
)
/
(
q
-
1
)
(
2
t
+
n
-
2
)
d
t
]
1
-
1
/
q
mm
×
[
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
(
1
-
t
)
r
(
2
t
+
n
-
2
)
m
m
m
m
×
|
f
(
n
)
(
b
)
|
q
(
1
-
t
)
s
|
f
(
n
)
(
a
)
|
q
t
s
d
t
∫
(
x
-
b
)
/
(
η
(
a
,
b
)
)
1
‍
]
1
/
q
[
∫
0
(
x
-
b
)
/
(
η
(
a
,
b
)
)
‍
t
(
n
q
-
q
-
r
)
/
(
q
-
1
)
(
n
-
2
t
)
d
t
]
1
-
1
/
q
}
.
The rest is also similar to the proof of Theorem22.Corollary 27.
Under the assumptions of Theorem26,
(1) ifr
=
0, then
(47)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
(
2
-
q
)
/
q
2
n
!
[
q
-
1
(
n
q
-
1
)
(
n
q
+
q
-
2
)
]
1
-
1
/
q
×
{
[
H
(
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
(
x
-
b
)
(
n
q
-
1
)
/
q
mm
×
[
n
(
n
q
+
q
-
2
)
η
(
a
,
b
)
-
2
(
n
q
-
1
)
(
x
-
b
)
]
1
-
1
/
q
mm
×
[
H
(
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
δ
|
f
(
n
)
(
b
)
|
θ
mm
×
[
(
n
-
1
)
(
n
q
-
2
)
η
(
a
,
b
)
-
2
(
n
q
-
1
)
(
b
-
x
)
]
1
-
1
/
q
mm
×
(
b
+
η
(
a
,
b
)
-
x
)
n
q
-
1
/
q
mm
×
[
H
(
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
;
(2) ifr
=
(
n
-
1
)
q, then
(48)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
n
-
2
+
2
/
q
2
n
!
×
{
[
H
(
(
n
-
1
)
q
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
[
H
(
(
n
-
1
)
q
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
a
)
|
α
|
f
(
n
)
(
b
)
|
β
[
(
x
-
b
)
(
n
η
(
a
,
b
)
-
x
+
b
)
]
1
-
1
/
q
mm
×
[
H
(
(
n
-
1
)
q
+
1
;
ξ
s
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
δ
mm
×
|
f
(
n
)
(
b
)
|
θ
[
(
b
+
η
(
a
,
b
)
-
x
)
m
m
m
m
m
m
m
m
m
×
(
(
n
-
1
)
η
(
a
,
b
)
+
x
-
b
)
]
1
-
1
/
q
mm
×
[
H
(
(
n
-
1
)
q
+
1
;
ξ
-
s
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
;
(3) ifs
=
1, then
(49)
|
f
(
b
)
+
f
(
b
+
η
(
a
,
b
)
)
2
-
1
η
(
a
,
b
)
∫
b
b
+
η
(
a
,
b
)
‍
f
(
x
)
d
x
+
∑
k
=
2
n
‍
1
2
k
!
η
(
a
,
b
)
×
[
(
b
-
x
)
k
-
1
(
2
x
-
2
b
-
k
η
(
a
,
b
)
)
+
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
mm
×
(
2
b
+
2
η
(
a
,
b
)
-
2
x
-
k
η
(
a
,
b
)
)
(
b
+
η
(
a
,
b
)
-
x
)
k
-
1
]
f
(
k
-
1
)
(
x
)
∫
b
b
+
η
(
a
,
b
)
‍
|
≤
[
η
(
a
,
b
)
]
(
r
+
2
-
q
)
/
q
2
n
!
[
q
-
1
(
n
q
-
r
-
1
)
(
n
q
+
q
-
r
-
2
)
]
1
-
1
/
q
×
{
[
H
(
r
+
1
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
|
f
(
n
)
(
b
)
|
(
x
-
b
)
(
n
q
-
r
-
1
)
/
q
mm
×
[
n
(
n
q
+
q
-
r
-
2
)
η
(
a
,
b
)
m
m
m
m
-
2
(
n
q
-
r
-
1
)
(
x
-
b
)
]
1
-
1
/
q
mm
×
[
H
(
r
+
1
;
ξ
q
,
x
-
b
η
(
a
,
b
)
)
]
1
/
q
+
|
f
(
n
)
(
a
)
|
mm
×
[
n
(
n
q
+
q
-
r
-
2
)
η
(
a
,
b
)
m
m
m
m
-
2
(
n
q
-
r
-
1
)
(
b
+
η
(
a
,
b
)
-
x
)
]
1
-
1
/
q
mm
×
(
b
+
η
(
a
,
b
)
-
x
)
(
n
q
-
r
-
1
)
/
q
mm
×
[
H
(
r
+
1
;
ξ
-
q
,
b
+
η
(
a
,
b
)
-
x
η
(
a
,
b
)
)
]
1
/
q
}
.



---

*Source: 725987-2014-07-20.xml*