# Generalizations on Some Hermite-Hadamard Type Inequalities for Differentiable Convex Functions with Applications to Weighted Means

Authors: Banyat Sroysang
Journal: The Scientific World Journal
 (2014)

---

## Abstract

Some new Hermite-Hadamard type inequalities for differentiable convex functions were presented by Xi and Qi. In this paper, we present new generalizations on the Xi-Qi inequalities.


---

## Body

## 1. Introduction

The Hermite-Hadamard inequality [1–3] states that if f is a convex function on [
a
,
b
], then
(1)
f
(
a
+
b
2
)
≤
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
≤
f
(
a
)
+
f
(
b
)
2
.Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Below we recall some Hermite-Hadamard type inequalities.In 1998, Dragomir and Agarwal [4] showed that (i) if |
f
′
(
x
)
| is convex on [
a
,
b
], then
(2)
|
f
(
a
)
+
f
(
b
)
2
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
(
b
-
a
)
(
|
f
′
(
a
)
|
+
|
f
′
(
b
)
|
)
8
,and (ii) if|
f
′
(
x
)
|
p
/
(
p
-
1
) is convex on [
a
,
b
] with p
>
1, then
(3)
|
f
(
a
)
+
f
(
b
)
2
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
2
(
p
+
1
)
1
/
p
×
(
|
f
′
(
a
)
|
p
/
(
p
-
1
)
+
|
f
′
(
b
)
|
p
/
(
p
-
1
)
2
)
(
p
-
1
)
/
p
.In 2000, Pearce and Pečarić [5] showed that if |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then(4)
|
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
,
|
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
4
(
|
f
′
(
a
)
|
q
+
|
f
′
(
b
)
|
q
2
)
1
/
q
.In 2004, Kirmaci [6] showed that if |
f
′
(
x
)
|
p
/
(
p
-
1
) is convex on [
a
,
b
] with p
>
1, then(5)
|
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
4
(
4
p
+
1
)
1
/
p
(
|
f
′
(
a
)
|
+
|
f
′
(
b
)
|
)
,
|
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
)
)
(
p
-
1
)
/
p
+
b
-
a
16
(
4
p
+
1
)
1
/
p
×
(
3
|
f
′
(
a
)
|
p
/
(
p
-
1
)
+
|
f
′
(
b
)
|
p
/
(
p
-
1
)
)
(
p
-
1
)
/
p
.In 2010, Sarikaya et al. [7] showed that if f
′
∈
L
[
a
,
b
] and |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(6)
|
1
6
[
f
(
a
)
+
f
(
b
)
+
4
f
(
a
+
b
2
)
]
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
12
(
2
q
+
1
+
1
3
(
q
+
1
)
)
1
/
q
×
[
(
3
|
f
′
(
a
)
|
q
+
|
f
′
(
b
)
|
q
4
)
1
/
q
+
(
|
f
′
(
a
)
|
q
+
3
|
f
′
(
b
)
|
q
4
)
1
/
q
]
,
|
1
6
[
f
(
a
)
+
f
(
b
)
+
4
f
(
a
+
b
2
)
]
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
5
(
b
-
a
)
72
×
[
(
61
|
f
′
(
a
)
|
q
+
29
|
f
′
(
b
)
|
q
90
)
1
/
q
+
(
29
|
f
′
(
a
)
|
q
+
61
|
f
′
(
b
)
|
q
90
)
1
/
q
]
.In 2012, Xi and Qi [8] showed that if λ, μ
∈
[
0,1
] and if f
′
∈
L
[
a
,
b
] and |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(7)
|
λ
f
(
a
)
+
μ
f
(
b
)
2
+
2
-
λ
-
μ
2
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
8
(
1
-
2
λ
+
2
λ
2
)
1
-
1
/
q
×
[
(
(
4
-
9
λ
+
12
λ
2
-
2
λ
3
)
|
f
′
(
a
)
|
q
+
(
2
-
3
λ
+
2
λ
3
)
|
f
′
(
b
)
|
q
)
×
(
6
)
-
1
]
1
/
q
+
b
-
a
8
(
1
-
2
μ
+
2
μ
2
)
1
-
1
/
q
×
[
(
(
2
-
3
μ
+
2
μ
3
)
|
f
′
(
a
)
|
q
+
(
4
-
9
μ
+
12
μ
2
-
2
μ
3
)
|
f
′
(
b
)
|
q
)
×
(
6
)
-
1
]
1
/
q
,
|
λ
f
(
a
)
+
μ
f
(
b
)
2
+
2
-
λ
-
μ
2
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
4
(
1
2
(
q
+
1
)
(
q
+
2
)
)
1
/
q
×
[
(
(
q
+
3
-
λ
)
(
1
-
λ
)
q
+
1
+
(
2
q
+
4
-
λ
)
λ
q
+
1
)
|
f
′
(
a
)
|
q
+
(
(
q
+
1
+
λ
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
)
|
f
′
(
b
)
|
q
]
1
/
q
+
b
-
a
4
(
1
2
(
q
+
1
)
(
q
+
2
)
)
1
/
q
×
[
(
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
+
μ
q
+
2
)
|
f
′
(
a
)
|
q
+
(
(
q
+
3
-
μ
)
(
1
-
μ
)
q
+
1
+
(
2
q
+
4
-
μ
)
μ
q
+
1
)
×
|
f
′
(
b
)
|
q
]
1
/
q
.Moreover, for other results involving the Hermite-Hadamard type inequalities, we also refer to [9–23].In this paper, we generalize the Xi-Qi inequalities.


## 2. Preliminaries

Lemma 1.
Letλ,  μ
∈
ℝ and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. Then
(8)
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
=
∫
0
1
‍
[
ϵ
2
(
1
-
λ
-
t
)
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
+
(
b
-
a
-
ϵ
)
2
(
μ
-
t
)
]
×
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
d
t
.Proof.
Integrating by part and changing variable, we have(9)
∫
0
1
‍
ϵ
2
(
1
-
λ
-
t
)
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
d
t
=
∫
t
=
0
t
=
1
‍
ϵ
(
λ
+
t
-
1
)
d
f
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
=
[
ϵ
(
λ
+
t
-
1
)
f
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
]
t
=
0
t
=
1
-
ϵ
∫
0
1
‍
f
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
d
t
=
ϵ
λ
f
(
a
)
+
ϵ
(
1
-
λ
)
f
(
a
+
ϵ
)
-
∫
a
a
+
ϵ
‍
f
(
x
)
d
x
,
∫
0
1
‍
(
b
-
a
-
ϵ
)
2
(
μ
-
t
)
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
d
t
=
∫
t
=
0
t
=
1
‍
(
b
-
a
-
ϵ
)
(
t
-
μ
)
d
f
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
=
[
(
b
-
a
-
ϵ
)
(
t
-
μ
)
f
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
]
t
=
0
t
=
1
-
(
b
-
a
-
ϵ
)
∫
0
1
‍
f
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
d
t
=
(
b
-
a
-
ϵ
)
(
1
-
μ
)
f
(
a
+
ϵ
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
-
∫
a
+
ϵ
b
‍
f
(
x
)
d
x
.
Thus,(10)
∫
0
1
‍
[
ϵ
2
(
1
-
λ
-
t
)
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
+
(
b
-
a
-
ϵ
)
2
(
μ
-
t
)
f
′
]
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
d
t
=
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
.Lemma 2 (see [8]).
Lets
>
0 and 0
≤
ξ
≤
1. Then
(11)
∫
0
1
‍
|
ξ
-
t
|
s
d
t
=
ξ
s
+
1
+
(
1
-
ξ
)
s
+
1
s
+
1
,
∫
0
1
‍
t
|
ξ
-
t
|
s
d
t
=
ξ
s
+
2
+
(
s
+
1
+
ξ
)
(
1
-
ξ
)
s
+
1
(
s
+
1
)
(
s
+
2
)
.


## 3. Main Results

Theorem 3.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(12)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
(
1
-
λ
)
2
+
λ
2
2
)
1
-
(
1
/
q
)
×
{
1
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
a
)
|
q
+
1
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
q
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
μ
2
+
(
1
-
μ
)
2
2
)
1
-
(
1
/
q
)
×
{
1
6
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
q
+
1
6
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
×
|
f
′
(
b
)
|
q
1
6
}
1
/
q
.Proof.
Suppose that|
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1. By Lemma 1, we have
(13)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
∫
0
1
‍
|
1
-
λ
-
t
|
×
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
d
t
+
(
b
-
a
-
ϵ
)
2
∫
0
1
‍
|
μ
-
t
|
×
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
d
t
.
Case  (
q
=
1
). By the convexity of |
f
′
(
x
)
| and Lemma 2, we have
(14)
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
d
t
=
∫
0
1
‍
|
1
-
λ
-
t
|
×
|
f
′
(
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
a
+
ϵ
(
1
-
t
)
b
-
a
b
)
|
d
t
≤
∫
0
1
‍
|
1
-
λ
-
t
|
×
[
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
|
f
′
(
a
)
|
+
ϵ
(
1
-
t
)
b
-
a
|
f
′
(
b
)
|
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
+
ϵ
b
-
a
|
f
′
(
b
)
|
]
×
∫
0
1
‍
|
1
-
λ
-
t
|
d
t
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
-
ϵ
b
-
a
|
f
′
(
b
)
|
]
×
∫
0
1
‍
t
|
1
-
λ
-
t
|
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
+
ϵ
b
-
a
|
f
′
(
b
)
|
]
×
(
1
-
λ
)
2
+
λ
2
2
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
-
ϵ
b
-
a
|
f
′
(
b
)
|
]
×
(
1
-
λ
)
3
+
(
3
-
λ
)
λ
2
6
=
1
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
a
)
|
+
1
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
,
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
d
t
=
∫
0
1
‍
|
μ
-
t
|
×
|
f
′
(
(
t
-
ϵ
t
b
-
a
)
a
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
b
)
|
d
t
≤
∫
0
1
‍
|
μ
-
t
|
[
(
t
-
ϵ
t
b
-
a
)
|
f
′
(
a
)
|
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
|
f
′
(
b
)
|
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
]
×
∫
0
1
‍
t
|
μ
-
t
|
d
t
+
|
f
′
(
b
)
|
∫
0
1
‍
|
μ
-
t
|
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
]
×
μ
3
+
(
2
+
μ
)
(
1
-
μ
)
2
6
+
|
f
′
(
b
)
|
μ
2
+
(
1
-
μ
)
2
2
=
1
6
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
+
1
6
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
×
|
f
′
(
b
)
|
.
Thus,
(15)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
a
)
|
+
ϵ
2
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
+
(
b
-
a
-
ϵ
)
2
6
×
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
+
(
b
-
a
-
ϵ
)
2
6
×
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
b
)
|
.
Case  (
q
>
1
). By Hölder's inequality, we have
(16)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
∫
0
1
‍
|
1
-
λ
-
t
|
d
t
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
∫
0
1
‍
|
μ
-
t
|
d
t
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
)
1
/
q
=
ϵ
2
(
(
1
-
λ
)
2
+
λ
2
2
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
μ
2
+
(
1
-
μ
)
2
2
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
)
1
/
q
.
By the convexity of|
f
′
(
x
)
|
q and Lemma 2, we have
(17)
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
=
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
a
+
ϵ
(
1
-
t
)
b
-
a
b
)
|
q
d
t
≤
∫
0
1
‍
|
1
-
λ
-
t
|
×
[
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
(
1
-
t
)
b
-
a
|
f
′
(
b
)
|
q
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
∫
0
1
‍
|
1
-
λ
-
t
|
d
t
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
q
-
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
∫
0
1
‍
t
|
1
-
λ
-
t
|
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
(
1
-
λ
)
2
+
λ
2
2
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
q
-
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
(
1
-
λ
)
3
+
(
3
-
λ
)
λ
2
6
=
1
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
a
)
|
q
+
1
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
q
,
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
=
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
(
t
-
ϵ
t
b
-
a
)
a
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
b
)
|
q
d
t
≤
∫
0
1
‍
|
μ
-
t
|
×
[
(
t
-
ϵ
t
b
-
a
)
|
f
′
(
a
)
|
q
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
|
f
′
(
b
)
|
q
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
q
]
×
∫
0
1
‍
t
|
μ
-
t
|
d
t
+
|
f
′
(
b
)
|
q
∫
0
1
‍
|
μ
-
t
|
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
q
]
×
μ
3
+
(
2
+
μ
)
(
1
-
μ
)
2
6
+
|
f
′
(
b
)
|
q
(
μ
2
+
(
1
-
μ
)
2
2
)
=
1
6
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
q
+
1
6
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
×
|
f
′
(
b
)
|
q
.
Thus,
(18)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
(
1
-
λ
)
2
+
λ
2
2
)
1
-
(
1
/
q
)
×
{
1
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
×
|
f
′
(
a
)
|
q
+
1
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
q
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
μ
2
+
(
1
-
μ
)
2
2
)
1
-
(
1
/
q
)
×
{
1
6
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
q
+
1
6
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
×
|
f
′
(
b
)
|
q
1
6
}
1
/
q
.
This proof is completed.It is easy to notice that if we putϵ
=
(
b
-
a
)
/
2 in Theorem 3 then we get the following.Corollary 4 (see [8]).
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(19)
|
λ
f
(
a
)
+
μ
f
(
b
)
2
+
2
-
λ
-
μ
2
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
8
(
1
-
2
λ
+
2
λ
2
)
1
-
1
/
q
×
[
(
(
4
-
9
λ
+
12
λ
2
-
2
λ
3
)
|
f
′
(
a
)
|
q
+
(
2
-
3
λ
+
2
λ
3
)
|
f
′
(
b
)
|
q
)
×
(
6
)
-
1
]
1
/
q
+
b
-
a
8
(
1
-
2
μ
+
2
μ
2
)
1
-
1
/
q
×
[
(
(
2
-
3
μ
+
2
μ
3
)
|
f
′
(
a
)
|
q
+
(
4
-
9
μ
+
12
μ
2
-
2
μ
3
)
|
f
′
(
b
)
|
q
)
×
(
6
)
-
1
]
1
/
q
.One can easily check that if we putϵ
=
(
b
-
a
)
/
3 in Theorem 3, then we get the following.Corollary 5.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(20)
|
∫
a
b
‍
λ
f
(
a
)
+
2
μ
f
(
b
)
3
+
(
1
-
λ
3
-
2
μ
3
)
f
(
2
a
+
b
3
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
9
(
(
1
-
λ
)
2
+
λ
2
2
)
1
-
(
1
/
q
)
×
(
1
18
(
7
-
15
λ
+
18
λ
2
-
2
λ
3
)
|
f
′
(
a
)
|
q
+
1
18
(
2
-
3
λ
+
2
λ
3
)
|
f
′
(
b
)
|
q
)
1
/
q
+
4
(
b
-
a
)
9
(
μ
2
+
(
1
-
μ
)
2
2
)
1
-
(
1
/
q
)
×
(
1
9
(
2
-
3
μ
+
2
μ
3
)
|
f
′
(
a
)
|
q
+
1
18
(
5
-
12
μ
+
18
μ
2
-
4
μ
3
)
|
f
′
(
b
)
|
q
)
1
/
q
.One can easily check that if we putϵ
=
2
(
b
-
a
)
/
3 in Theorem 3 then we get the following.Corollary 6.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(21)
|
∫
a
b
‍
2
λ
f
(
a
)
+
μ
f
(
b
)
3
+
(
1
-
2
λ
3
-
μ
3
)
f
(
a
+
2
b
3
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
4
(
b
-
a
)
9
(
(
1
-
λ
)
2
+
λ
2
2
)
1
-
(
1
/
q
)
×
(
1
18
(
5
-
12
λ
+
18
λ
2
-
4
λ
3
)
|
f
′
(
a
)
|
q
+
1
9
(
2
-
3
λ
+
2
λ
3
)
|
f
′
(
b
)
|
q
)
1
/
q
+
b
-
a
9
(
μ
2
+
(
1
-
μ
)
2
2
)
1
-
(
1
/
q
)
×
(
1
18
(
2
-
3
μ
+
2
μ
3
)
|
f
′
(
a
)
|
q
+
1
18
(
7
-
15
μ
+
18
μ
2
-
2
μ
3
)
|
f
′
(
b
)
|
q
)
1
/
q
.It is easy to notice that if we putλ
=
μ
=
0 in Theorem 3 then we get the following.Corollary 7.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(22)
2
1
-
(
1
/
q
)
(
b
-
a
)
|
f
(
a
+
ϵ
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
(
1
2
-
ϵ
3
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
ϵ
3
(
b
-
a
)
|
f
′
(
b
)
|
q
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
(
1
3
-
ϵ
3
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
(
1
6
+
ϵ
3
(
b
-
a
)
)
|
f
′
(
b
)
|
q
)
1
/
q
.It is easy to notice that if we putλ
=
μ
=
1
/
2 in Theorem 3 then we get the following.Corollary 8.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L  [
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(23)
4
1
-
(
1
/
q
)
|
∫
a
b
‍
ϵ
f
(
a
)
+
(
b
-
a
-
ϵ
)
f
(
b
)
+
(
b
-
a
)
f
(
a
+
ϵ
)
2
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
(
1
4
-
ϵ
8
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
ϵ
8
(
b
-
a
)
|
f
′
(
b
)
|
q
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
(
1
8
-
ϵ
8
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
(
1
8
+
ϵ
8
(
b
-
a
)
)
|
f
′
(
b
)
|
q
)
1
/
q
.It is easy to notice that if we putλ
=
μ
=
1 in Theorem 3 then we get the following.Corollary 9.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(24)
2
1
-
(
1
/
q
)
|
ϵ
f
(
a
)
+
(
b
-
a
-
ϵ
)
f
(
b
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
(
1
3
-
ϵ
6
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
ϵ
6
(
b
-
a
)
|
f
′
(
b
)
|
q
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
(
1
6
-
ϵ
6
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
(
1
3
+
ϵ
6
(
b
-
a
)
)
|
f
′
(
b
)
|
q
)
1
/
q
.Theorem 10.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(25)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
{
[
(
1
-
λ
)
q
+
1
+
λ
q
+
1
q
+
1
-
(
ϵ
b
-
a
)
×
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
a
)
|
q
+
[
(
ϵ
b
-
a
)
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
b
)
|
q
[
(
ϵ
b
-
a
)
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
{
[
(
1
-
ϵ
b
-
a
)
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
a
)
|
q
+
[
μ
q
+
1
+
(
1
-
μ
)
q
+
1
q
+
1
-
(
1
-
ϵ
b
-
a
)
×
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
b
)
|
q
[
(
1
-
ϵ
b
-
a
)
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
}
1
/
q
.Proof.
Suppose that|
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1. If q
=
1, then, by Theorem 3, we have(26)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
{
1
6
[
3
-
6
λ
+
6
λ
2
-
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
a
)
|
+
1
6
[
ϵ
b
-
a
(
2
-
3
λ
+
2
λ
3
)
]
|
f
′
(
b
)
|
}
+
(
b
-
a
-
ϵ
)
2
×
{
1
6
[
(
1
-
ϵ
b
-
a
)
(
2
-
3
μ
+
2
μ
3
)
]
|
f
′
(
a
)
|
+
1
6
[
1
-
3
μ
+
6
μ
2
-
2
μ
3
+
ϵ
b
-
a
(
2
-
3
μ
+
2
μ
3
)
]
×
|
f
′
(
b
)
|
1
6
}
=
ϵ
2
{
[
(
1
-
λ
)
2
+
λ
2
2
-
(
ϵ
b
-
a
)
(
2
+
λ
)
(
1
-
λ
)
2
+
λ
3
6
]
×
|
f
′
(
a
)
|
+
[
(
ϵ
b
-
a
)
(
2
+
λ
)
(
1
-
λ
)
2
+
λ
3
6
]
×
|
f
′
(
b
)
|
[
(
ϵ
b
-
a
)
(
2
+
λ
)
(
1
-
λ
)
2
+
λ
3
6
]
}
+
(
b
-
a
-
ϵ
)
2
×
{
[
(
1
-
ϵ
b
-
a
)
μ
3
+
(
2
+
μ
)
(
1
-
μ
)
2
6
]
|
f
′
(
a
)
|
+
[
μ
2
+
(
1
-
μ
)
2
2
-
(
1
-
ϵ
b
-
a
)
×
μ
3
+
(
2
+
μ
)
(
1
-
μ
)
2
6
]
|
f
′
(
b
)
|
}
.
Next, we suppose thatq
>
1. By Lemma 1 and Hölder’s inequality, we have
(27)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
∫
0
1
‍
|
1
-
λ
-
t
|
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
d
t
+
(
b
-
a
-
ϵ
)
2
∫
0
1
‍
|
μ
-
t
|
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
d
t
≤
ϵ
2
(
∫
0
1
‍
d
t
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
1
-
λ
-
t
|
q
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
∫
0
1
‍
d
t
)
1
-
(
1
/
q
)
×
(
∫
0
1
‍
|
μ
-
t
|
q
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
)
1
/
q
=
ϵ
2
(
∫
0
1
‍
|
1
-
λ
-
t
|
q
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
∫
0
1
‍
|
μ
-
t
|
q
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
)
1
/
q
.
By the convexity of|
f
′
(
x
)
|
q and Lemma 2, we have
(28)
∫
0
1
‍
|
1
-
λ
-
t
|
q
|
f
′
(
t
a
+
(
1
-
t
)
(
a
+
ϵ
)
)
|
q
d
t
=
∫
0
1
‍
|
1
-
λ
-
t
|
q
|
f
′
(
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
a
+
ϵ
(
1
-
t
)
b
-
a
b
)
|
q
d
t
≤
∫
0
1
‍
|
1
-
λ
-
t
|
q
×
[
(
1
-
ϵ
(
1
-
t
)
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
(
1
-
t
)
b
-
a
|
f
′
(
b
)
|
q
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
×
∫
0
1
‍
|
1
-
λ
-
t
|
q
d
t
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
q
-
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
×
∫
0
1
‍
t
|
1
-
λ
-
t
|
q
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
+
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
×
(
1
-
λ
)
q
+
1
+
λ
q
+
1
q
+
1
+
[
ϵ
b
-
a
|
f
′
(
a
)
|
q
-
ϵ
b
-
a
|
f
′
(
b
)
|
q
]
×
(
1
-
λ
)
q
+
2
+
(
q
+
2
-
λ
)
λ
q
+
1
(
q
+
1
)
(
q
+
2
)
=
[
(
1
-
λ
)
q
+
1
+
λ
q
+
1
q
+
1
-
(
ϵ
b
-
a
)
×
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
a
)
|
q
+
[
(
ϵ
b
-
a
)
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
b
)
|
q
,
∫
0
1
‍
|
μ
-
t
|
q
|
f
′
(
t
(
a
+
ϵ
)
+
(
1
-
t
)
b
)
|
q
d
t
=
∫
0
1
‍
|
μ
-
t
|
q
×
|
f
′
(
(
t
-
ϵ
t
b
-
a
)
a
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
b
)
|
q
d
t
≤
∫
0
1
‍
|
μ
-
t
|
q
×
[
(
t
-
ϵ
t
b
-
a
)
|
f
′
(
a
)
|
q
+
(
1
-
(
t
-
ϵ
t
b
-
a
)
)
|
f
′
(
b
)
|
q
]
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
q
]
×
∫
0
1
‍
t
|
μ
-
t
|
q
d
t
+
|
f
′
(
b
)
|
q
∫
0
1
‍
|
μ
-
t
|
q
d
t
=
[
(
1
-
ϵ
b
-
a
)
|
f
′
(
a
)
|
q
-
(
1
-
ϵ
b
-
a
)
|
f
′
(
b
)
|
q
]
×
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
+
|
f
′
(
b
)
|
q
(
μ
q
+
1
+
(
1
-
μ
)
q
+
1
q
+
1
)
=
[
(
1
-
ϵ
b
-
a
)
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
a
)
|
q
+
[
μ
q
+
1
+
(
1
-
μ
)
q
+
1
q
+
1
-
(
1
-
ϵ
b
-
a
)
×
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
b
)
|
q
.
Thus,
(29)
|
∫
a
b
‍
ϵ
λ
f
(
a
)
+
(
b
-
a
-
ϵ
)
μ
f
(
b
)
+
[
ϵ
(
1
-
λ
)
+
(
b
-
a
-
ϵ
)
(
1
-
μ
)
]
×
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
{
[
(
1
-
λ
)
q
+
1
+
λ
q
+
1
q
+
1
-
(
ϵ
b
-
a
)
×
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
|
f
′
(
a
)
|
q
+
[
(
ϵ
b
-
a
)
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
b
)
|
q
[
(
ϵ
b
-
a
)
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
(
q
+
1
)
(
q
+
2
)
]
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
{
[
(
1
-
ϵ
b
-
a
)
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
a
)
|
q
+
[
μ
q
+
1
+
(
1
-
μ
)
q
+
1
q
+
1
-
(
1
-
ϵ
b
-
a
)
×
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
×
|
f
′
(
b
)
|
q
[
(
1
-
ϵ
b
-
a
)
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
(
q
+
1
)
(
q
+
2
)
]
}
1
/
q
.
This proof is completed.It is easy to notice that if we putϵ
=
(
b
-
a
)
/
2 in Theorem 10 then we get the following.Corollary 11 (see [8]).
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(30)
|
λ
f
(
a
)
+
μ
f
(
b
)
2
+
2
-
λ
-
μ
2
f
(
a
+
b
2
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
4
(
1
2
(
q
+
1
)
(
q
+
2
)
)
1
/
q
×
[
(
(
q
+
3
-
λ
)
(
1
-
λ
)
q
+
1
+
(
2
q
+
4
-
λ
)
λ
q
+
1
)
|
f
′
(
a
)
|
q
+
(
(
q
+
1
+
λ
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
)
|
f
′
(
b
)
|
q
]
1
/
q
+
b
-
a
4
(
1
2
(
q
+
1
)
(
q
+
2
)
)
1
/
q
×
[
(
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
+
μ
q
+
2
)
|
f
′
(
a
)
|
q
+
(
(
q
+
3
-
μ
)
(
1
-
μ
)
q
+
1
+
(
2
q
+
4
-
μ
)
μ
q
+
1
)
×
|
f
′
(
b
)
|
q
]
1
/
q
.One can easily check that if we putϵ
=
(
b
-
a
)
/
3 in Theorem 10 then we get the following.Corollary 12.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(31)
|
∫
a
b
‍
λ
f
(
a
)
+
2
μ
f
(
b
)
3
+
(
1
-
λ
3
-
2
μ
3
)
f
(
2
a
+
b
3
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
b
-
a
9
×
{
(
5
+
2
q
-
λ
)
(
1
-
λ
)
q
+
1
+
(
6
+
3
q
-
λ
)
λ
q
+
1
3
(
q
+
1
)
(
q
+
2
)
|
f
′
(
a
)
|
q
+
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
λ
q
+
2
3
(
q
+
1
)
(
q
+
2
)
|
f
′
(
b
)
|
q
}
1
/
q
+
4
(
b
-
a
)
9
×
{
2
μ
q
+
2
+
2
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
|
f
′
(
a
)
|
q
+
(
6
+
3
q
-
2
μ
)
μ
q
+
1
+
(
4
+
q
-
2
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
×
|
f
′
(
b
)
|
q
2
μ
q
+
2
+
2
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
}
1
/
q
.One can easily check that if we putϵ
=
2
(
b
-
a
)
/
3 in Theorem 10 then we get the following.Corollary 13.
Letλ,  μ
∈
[
0,1
] and let f
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L  [
a
,
b
]. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(32)
|
∫
a
b
‍
2
λ
f
(
a
)
+
μ
f
(
b
)
3
+
(
1
-
2
λ
3
-
μ
3
)
f
(
a
+
2
b
3
)
-
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
|
≤
4
(
b
-
a
)
9
×
{
(
4
-
q
-
2
λ
)
(
1
-
λ
)
q
+
1
+
(
6
+
3
q
-
2
λ
)
λ
q
+
1
3
(
q
+
1
)
(
q
+
2
)
×
|
f
′
(
a
)
|
q
+
2
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
2
λ
q
+
2
3
(
q
+
1
)
(
q
+
2
)
×
|
f
′
(
b
)
|
q
2
(
1
+
λ
+
q
)
(
1
-
λ
)
q
+
1
+
2
λ
q
+
2
3
(
q
+
1
)
(
q
+
2
)
}
1
/
q
+
b
-
a
9
×
{
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
|
f
′
(
a
)
|
q
+
(
6
+
3
q
-
μ
)
μ
q
+
1
+
(
5
+
2
q
-
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
×
|
f
′
(
b
)
|
q
μ
q
+
2
+
(
q
+
1
+
μ
)
(
1
-
μ
)
q
+
1
3
(
q
+
1
)
(
q
+
2
)
}
1
/
q
.It is easy to notice that if we putλ
=
μ
=
0 in Theorem 10 then we get the following.Corollary 14.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(33)
|
(
b
-
a
)
f
(
a
+
ϵ
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
{
(
1
q
+
1
-
ϵ
(
b
-
a
)
(
q
+
2
)
)
|
f
′
(
a
)
|
q
+
ϵ
(
b
-
a
)
(
q
+
2
)
|
f
′
(
b
)
|
q
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
{
(
1
-
ϵ
b
-
a
)
(
1
q
+
2
)
|
f
′
(
a
)
|
q
+
(
1
q
+
1
-
(
1
-
ϵ
b
-
a
)
(
1
q
+
2
)
)
|
f
′
(
b
)
|
q
}
1
/
q
.It is easy to notice that if we putλ
=
μ
=
1
/
2 in Theorem 10 then we get the following.Corollary 15.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(34)
|
ϵ
f
(
a
)
+
(
b
-
a
-
ϵ
)
f
(
b
)
+
(
b
-
a
)
f
(
a
+
ϵ
)
2
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
2
(
1
q
+
1
)
1
/
q
{
(
1
-
ϵ
2
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
ϵ
2
(
b
-
a
)
|
f
′
(
b
)
|
q
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
2
(
1
q
+
1
)
1
/
q
×
{
(
1
2
-
ϵ
2
(
b
-
a
)
)
|
f
′
(
a
)
|
q
+
(
1
2
+
ϵ
2
(
b
-
a
)
)
|
f
′
(
b
)
|
q
}
1
/
q
.It is easy to notice that if we putλ
=
μ
=
1 in Theorem 10 then we get the following.Corollary 16.
Letf
:
I
⊆
ℝ
→
ℝ be differentiable on I
∘ and {
a
,
b
}
⊆
I with a
<
b. Assume that f
′
∈
L
[
a
,
b
] and 0
<
ϵ
<
b
-
a. If |
f
′
(
x
)
|
q is convex on [
a
,
b
] with q
≥
1, then
(35)
|
ϵ
f
(
a
)
+
(
b
-
a
-
ϵ
)
f
(
b
)
-
∫
a
b
‍
f
(
x
)
d
x
|
≤
ϵ
2
(
1
q
+
1
)
1
/
q
{
(
1
-
ϵ
(
b
-
a
)
(
q
+
2
)
)
|
f
′
(
a
)
|
q
+
ϵ
(
b
-
a
)
(
q
+
2
)
|
f
′
(
b
)
|
q
}
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
1
q
+
1
)
1
/
q
×
{
(
1
-
ϵ
b
-
a
)
(
1
q
+
2
)
|
f
′
(
a
)
|
q
+
(
1
-
(
1
-
ϵ
b
-
a
)
(
1
q
+
2
)
)
|
f
′
(
b
)
|
q
}
1
/
q
.


## 4. Applications

In this section, we suppose that{
s
,
q
}
⊆
[
1
,
∞
) and {
a
,
b
,
w
a
,
w
b
}
⊆
(
0
,
∞
) with a
<
b. Let 0
<
ϵ
<
b
-
a.The weighted arithmetic mean of data{
a
,
b
} with weight {
w
a
,
w
b
} is defined by
(36)
A
(
a
,
b
;
w
a
,
w
b
)
=
w
a
a
+
w
b
b
w
a
+
w
b
.The weighted geometric mean of data{
a
,
b
} with weight {
w
a
,
w
b
} is defined by
(37)
G
(
a
,
b
;
w
a
,
w
b
)
=
a
w
a
/
(
w
a
+
w
b
)
b
w
b
/
(
w
a
+
w
b
)
.The generalized logarithmic mean of data{
a
,
b
} is defined by
(38)
L
(
a
,
b
)
=
(
b
s
+
1
-
a
s
+
1
(
s
+
1
)
(
b
-
a
)
)
1
/
s
.The identric mean of data{
a
,
b
} is defined by
(39)
I
(
a
,
b
)
=
1
e
(
b
b
a
a
)
1
/
(
b
-
a
)
.Applying Corollary7 with f
(
x
)
=
x
s on (
0
,
∞
), we get the following:
(40)
2
1
-
(
1
/
q
)
(
b
-
a
)
s
|
A
s
(
a
,
b
;
b
-
a
-
ϵ
,
ϵ
)
-
L
s
(
a
,
b
)
|
≤
ϵ
2
(
a
(
s
-
1
)
q
2
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
3
(
b
-
a
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
2
a
(
s
-
1
)
q
+
b
(
s
-
1
)
q
6
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
3
(
b
-
a
)
)
1
/
q
.Applying Corollary9 with f
(
x
)
=
x
s on (
0
,
∞
), we get the following:
(41)
2
1
-
(
1
/
q
)
(
b
-
a
)
s
|
A
(
a
s
,
b
s
;
ϵ
,
b
-
a
-
ϵ
)
-
L
s
(
a
,
b
)
|
≤
ϵ
2
(
a
(
s
-
1
)
q
3
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
6
(
b
-
a
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
a
(
s
-
1
)
q
+
2
b
(
s
-
1
)
q
6
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
6
(
b
-
a
)
)
1
/
q
.Applying Corollary7 with f
(
x
)
=
ln
⁡
x on (
0
,
∞
), we get the following:
(42)
2
1
-
(
1
/
q
)
(
b
-
a
)
|
ln
⁡
(
A
(
a
,
b
;
b
-
a
-
ϵ
,
ϵ
)
I
(
a
,
b
)
)
|
≤
ϵ
2
(
a
-
q
2
+
ϵ
(
b
-
q
-
a
-
q
)
3
(
b
-
a
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
2
a
-
q
+
b
-
q
6
+
ϵ
(
b
-
q
-
a
-
q
)
3
(
b
-
a
)
)
1
/
q
.Applying Corollary9 with f
(
x
)
=
ln
⁡
x on (
0
,
∞
), we get the following:
(43)
2
1
-
(
1
/
q
)
(
b
-
a
)
|
ln
⁡
(
G
(
a
,
b
;
ϵ
,
b
-
a
-
ϵ
)
I
(
a
,
b
)
)
|
≤
ϵ
2
(
a
-
q
3
+
ϵ
(
b
-
q
-
a
-
q
)
6
(
b
-
a
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
a
-
q
+
2
b
-
q
6
+
ϵ
(
b
-
q
-
a
-
q
)
6
(
b
-
a
)
)
1
/
q
.Applying Corollary14 with f
(
x
)
=
x
s on (
0
,
∞
), we get the following:
(44)
b
-
a
s
|
A
s
(
a
,
b
;
b
-
a
-
ϵ
,
ϵ
)
-
L
s
(
a
,
b
)
|
≤
ϵ
2
(
a
(
s
-
1
)
q
q
+
1
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
b
(
s
-
1
)
q
q
+
1
+
a
(
s
-
1
)
q
-
b
(
s
-
1
)
q
q
+
2
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
.Applying Corollary16 with f
(
x
)
=
x
s on (
0
,
∞
), we get the following:
(45)
(
q
+
1
)
1
/
q
(
b
-
a
)
s
|
A
(
a
s
,
b
s
;
ϵ
,
b
-
a
-
ϵ
)
-
L
s
(
a
,
b
)
|
≤
ϵ
2
(
a
(
s
-
1
)
q
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
×
(
b
(
s
-
1
)
q
+
a
(
s
-
1
)
q
-
b
(
s
-
1
)
q
q
+
2
+
ϵ
(
b
(
s
-
1
)
q
-
a
(
s
-
1
)
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
.Applying Corollary14 with f
(
x
)
=
ln
⁡
x on (
0
,
∞
), we get the following:
(46)
(
b
-
a
)
|
ln
⁡
(
A
(
a
,
b
;
b
-
a
-
ϵ
,
ϵ
)
I
(
a
,
b
)
)
|
≤
ϵ
2
(
a
-
q
q
+
1
+
ϵ
(
b
-
q
-
a
-
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
b
-
q
q
+
1
+
a
-
q
-
b
-
q
q
+
2
+
ϵ
(
b
-
q
-
a
-
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
.Applying Corollary16 with f
(
x
)
=
ln
⁡
x on (
0
,
∞
), we get the following:
(47)
(
q
+
1
)
1
/
q
(
b
-
a
)
|
ln
⁡
(
G
(
a
,
b
;
ϵ
,
b
-
a
-
ϵ
)
I
(
a
,
b
)
)
|
≤
ϵ
2
(
a
-
q
+
ϵ
(
b
-
q
-
a
-
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
+
(
b
-
a
-
ϵ
)
2
(
b
-
q
+
a
-
q
-
b
-
q
q
+
2
+
ϵ
(
b
-
q
-
a
-
q
)
(
b
-
a
)
(
q
+
2
)
)
1
/
q
.



---

*Source: 717164-2014-01-16.xml*