# On Some Integral Inequalities Related to Hermite-Hadamard-Fejér Inequalities for Coordinated Convex Functions

Authors: Ruiyin Xiang; Feixiang Chen
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

Several new mappings associated with coordinated convexity are proposed, by which we obtain some new Hermite-Hadamard-Fejér type inequalities for coordinated convex functions. We conclude that the results obtained in this work are the generalizations of the earlier results.


---

## Body

## 1. Introduction

Letf
:
I
⊆
R
→
R be a convex function and a
,
b
∈
I with a
<
b; then
(1)
f
a
+
b
2
≤
1
b
-
a
∫
a
b
‍
f
(
t
)
d
t
≤
f
(
a
)
+
f
(
b
)
2
is known as the Hermite-Hadamard inequality.In [1], Fejér established the following weighted generalization of inequality (1).Theorem 1.
Iff
:
[
a
,
b
]
→
R is a convex function, then the inequality
(2)
f
a
+
b
2
∫
a
b
‍
p
x
d
x
≤
∫
a
b
‍
f
(
x
)
p
(
x
)
d
x
≤
f
(
a
)
+
f
(
b
)
2
∫
a
b
‍
p
(
x
)
d
x
holds, where p
:
[
a
,
b
]
→
R is positive, integrable, and symmetric about x
=
a
+
b
/
2.Inequalities (1) and (2) have been extended, generalized, and improved by a number of authors (e.g., [2–9]).In [4], Dragomir proposed the following Hermite-Hadamard type inequalities which refine the first inequality of (1).Theorem 2 (see [4]).
Letf be convex on [
a
,
b
]. Then H is convex, increasing on [
0,1
], and, for all t
∈
[
0,1
],
(3)
f
a
+
b
2
=
H
(
0
)
≤
H
(
t
)
≤
H
(
1
)
=
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
,
where
(4)
H
(
t
)
=
1
b
-
a
∫
a
b
‍
f
t
s
+
1
-
t
a
+
b
2
d
s
.An analogous result for convex functions which refines the second inequality of (1) is obtained by Yang and Hong in [10] as follows.Theorem 3 (see [10]).
Letf be convex on [
a
,
b
]. Then P is convex, increasing on [
0,1
], and, for all t
∈
[
0,1
],
(5)
1
b
-
a
∫
a
b
‍
f
(
x
)
d
x
=
P
(
0
)
≤
P
(
t
)
≤
P
(
1
)
=
f
(
a
)
+
f
(
b
)
2
,
where
(6)
P
(
t
)
=
1
2
(
b
-
a
)
×
∫
a
b
‍
f
1
+
t
2
a
+
1
-
t
2
s
d
d
d
d
d
+
f
1
+
t
2
b
+
1
-
t
2
s
d
s
.A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].Let us consider the bidimensional intervalΔ
:
=
[
a
,
b
]
×
[
c
,
d
] in R
2 with a
<
b and c
<
d; a mapping f
:
Δ
→
R is said to be convex on Δ if the inequality
(7)
f
(
λ
x
+
(
1
-
λ
)
z
,
λ
y
+
(
1
-
λ
)
w
)
≤
λ
f
x
,
y
+
1
-
λ
f
z
,
w
holds for all (
x
,
y
), (
z
,
w
)
∈
Δ and λ
∈
[
0,1
].A functionf
:
Δ
→
R is said to be coordinated convex on Δ if the partial mappings f
y
:
[
a
,
b
]
→
R, f
y
(
u
)
=
f
(
u
,
y
), and f
x
:
[
c
,
d
]
→
R, f
x
(
v
)
=
f
(
x
,
v
) are convex for all y
∈
[
c
,
d
] and x
∈
[
a
,
b
].A formal definition for coordinated convex functions may be stated as follows.Definition 4.
A functionf
:
Δ
→
R is said to be convex on coordinates on Δ if the inequality
(8)
f
λ
x
+
1
-
λ
z
,
t
y
+
1
-
t
w
≤
λ
t
f
(
x
,
y
)
+
λ
1
-
t
f
x
,
w
+
(
1
-
λ
)
t
f
(
z
,
y
)
+
(
1
-
t
)
(
1
-
λ
)
f
(
z
,
w
)
holds for all (
x
,
y
), (
z
,
y
), (
x
,
w
), and (
z
,
w
)
∈
Δ and t
,
λ
∈
[
0,1
].Dragomir in [5] established the following Hermite-Hadamard type inequalities for coordinated convex functions in a rectangle from the plane R
2.Theorem 5.
Suppose thatf
:
Δ
=
[
a
,
b
]
×
[
c
,
d
]
→
R is convex on the coordinates on Δ. Then one has the following inequalities:
(9)
f
a
+
b
2
,
c
+
d
2
≤
1
b
-
a
d
-
c
∫
a
b
‍
∫
c
d
‍
f
x
,
y
d
y
d
x
≤
f
a
,
c
+
f
a
,
d
+
f
b
,
c
+
f
b
,
d
4
.The mapping connected with the first inequality of (9) is considered in [5].Iff
:
[
a
,
b
]
×
[
c
,
d
]
→
R is a coordinated convex function, then the following mapping on [
0,1
] can be defined by
(10)
H
(
t
,
s
)
=
1
(
b
-
a
)
(
d
-
c
)
×
∫
a
b
‍
∫
c
d
‍
f
t
x
+
1
-
t
a
+
b
2
,
s
y
+
1
-
s
c
+
d
2
d
y
d
x
.
The mapping H has the following properties:(i)
His coordinated convex and monotonic nondecreasing on [
0,1
]
2;
(ii)
we have the following bounds forH:
(11)
sup
⁡
(
t
,
s
)
∈
[
0,1
]
2
H
t
,
s
=
1
b
-
a
d
-
c
∫
a
b
‍
∫
c
d
‍
f
x
,
y
d
y
d
x
=
H
(
1,1
)
,
inf
⁡
(
t
,
s
)
∈
[
0,1
]
2
H
(
t
,
s
)
=
f
a
+
b
2
,
c
+
d
2
=
H
(
0,0
)
.Recently, Hwang et al. [11] established a monotonic nondecreasing mapping connected with the Hadamard’s inequality for coordinated convex functions in a rectangle from the plane as follows.Theorem 6 (see [11]).
Suppose thatf
:
Δ
=
[
a
,
b
]
×
[
c
,
d
]
→
R is coordinated convex on Δ and the mapping F
:
[
0,1
]
2
→
R is defined by
(12)
F
(
t
,
s
)
=
1
4
(
b
-
a
)
(
d
-
c
)
×
∫
a
b
‍
∫
c
d
‍
1
-
s
2
y
f
1
+
t
2
a
+
1
-
t
2
x
,
1
+
s
2
c
s
s
s
s
s
s
s
d
d
d
d
s
s
s
+
1
-
s
2
y
sssssssddsss
+
f
1
+
t
2
a
+
1
-
t
2
x
,
1
+
s
2
d
sssssssdsssddddd
+
1
-
s
2
y
sssssssddsss
+
f
1
+
t
2
b
+
1
-
t
2
x
,
1
+
s
2
c
sssssssddssddddd
+
1
-
s
2
y
sssssssddsss
+
f
1
+
t
2
b
+
1
-
t
2
x
,
1
+
s
2
d
sssssssddssddddd
+
1
-
s
2
y
d
y
d
x
.
Then(i)
the mappingF is coordinated convex on [
0,1
]
2;
(ii)
the mappingF is coordinated monotonic nondecreasing on [
0,1
]
2;
(iii)
we have the bounds(13)
inf
⁡
(
t
,
s
)
∈
[
0,1
]
2
F
t
,
s
=
1
b
-
a
d
-
c
∫
a
b
‍
∫
c
d
‍
f
x
,
y
d
y
d
x
=
F
(
0,0
)
,
sup
⁡
(
t
,
s
)
∈
[
0,1
]
2
F
t
,
s
=
f
(
a
,
c
)
+
f
(
a
,
d
)
+
f
(
b
,
c
)
+
f
(
b
,
d
)
4
=
F
(
1,1
)
.Fejér-type inequality for coordinated convex mappings is established in [12] as follows.Theorem 7 (see [12]).
Suppose thatf
:
Δ
=
[
a
,
b
]
×
[
c
,
d
]
→
R is a coordinated convex function on Δ. Then one has the following inequalities:
(14)
f
a
+
b
2
,
c
+
d
2
∫
a
b
‍
∫
c
d
‍
p
x
,
y
d
y
d
x
≤
∫
a
b
‍
∫
c
d
‍
f
x
,
y
p
x
,
y
d
y
d
x
≤
f
(
a
,
c
)
+
f
(
a
,
d
)
+
f
(
b
,
c
)
+
f
(
b
,
d
)
4
×
∫
a
b
‍
∫
c
d
‍
p
(
x
,
y
)
d
y
d
x
,
where p
:
[
a
,
b
]
×
[
c
,
d
]
→
R is positive, integrable, and symmetric about x
=
a
+
b
/
2 and y
=
c
+
d
/
2.Theorem 8 (see [12]).
Letf
:
[
a
,
b
]
×
[
c
,
d
]
→
R be a coordinated convex function, and consider the function P
:
[
a
,
b
]
×
[
c
,
d
]
→
R which is positive, integrable, and symmetric about x
=
a
+
b
/
2 and y
=
c
+
d
/
2. Let S be a function defined on [
0,1
]
2 by
(15)
S
t
,
s
=
1
(
b
-
a
)
(
d
-
c
)
×
∫
a
b
‍
∫
c
d
‍
f
t
x
+
1
-
t
a
+
b
2
,
s
y
+
1
-
s
c
+
d
2
d
d
d
d
d
d
×
p
x
,
y
d
y
d
x
.
Then S is a coordinated convex function on [
0,1
]
2, nondecreasing on [
0,1
]
2. Moreover,
(16)
inf
⁡
(
t
,
s
)
∈
[
0,1
]
2
S
t
,
s
=
S
(
0,0
)
=
f
a
+
b
2
,
c
+
d
2
∫
a
b
‍
∫
c
d
‍
p
x
,
y
d
y
d
x
,
sup
⁡
(
t
,
s
)
∈
[
0,1
]
2
S
(
t
,
s
)
=
S
(
1,1
)
=
∫
a
b
‍
∫
c
d
‍
f
x
,
y
p
x
,
y
d
y
d
x
.Theorem 9 (see [12]).
Letf
:
[
a
,
b
]
×
[
c
,
d
]
→
R be a coordinated convex function, and consider the function P
:
[
a
,
b
]
×
[
c
,
d
]
→
R which is positive, integrable, and symmetric about x
=
a
+
b
/
2 and y
=
c
+
d
/
2. Let A be a function defined on [
0,1
]
2 by
(17)
A
(
t
,
s
)
=
1
4
∫
a
b
‍
∫
c
d
‍
f
1
+
t
2
a
+
1
-
t
2
x
,
1
+
s
2
c
+
1
-
s
2
y
d
d
d
d
d
d
d
d
d
×
p
x
+
a
2
,
y
+
c
2
d
d
d
d
d
d
d
d
d
+
f
1
+
t
2
a
+
1
-
t
2
x
,
1
+
s
2
d
+
1
-
s
2
y
d
d
d
d
d
d
d
d
d
×
p
x
+
a
2
,
y
+
d
2
d
d
d
d
d
d
d
d
d
+
f
1
+
t
2
b
+
1
-
t
2
x
,
1
+
s
2
c
+
1
-
s
2
y
d
d
d
d
d
d
d
d
d
×
p
x
+
b
2
,
y
+
c
2
d
d
d
d
d
d
d
d
d
+
f
1
+
t
2
b
+
1
-
t
2
x
,
1
+
s
2
d
+
1
-
s
2
y
d
d
d
d
d
d
d
d
d
×
p
x
+
b
2
,
y
+
d
2
d
y
d
x
.
Then, A is coordinated convex on [
0,1
]
2, symmetric about (
1
/
2
,
1
/
2
), and non-decreasing on [
0,1
]
2, and
(18)
inf
⁡
(
t
,
s
)
∈
[
0,1
]
2
A
t
,
s
=
A
(
0,0
)
=
1
b
-
a
d
-
c
×
∫
a
b
‍
∫
c
d
‍
f
x
,
y
p
x
,
y
d
y
d
x
,
sup
⁡
(
t
,
s
)
∈
[
0,1
]
2
A
t
,
s
=
A
(
1,1
)
=
f
a
,
c
+
f
a
,
d
+
f
b
,
c
+
f
b
,
d
4
×
∫
a
b
‍
∫
c
d
‍
p
x
,
y
d
y
d
x
.In this paper, we establish some new results about the Hermite-Hadamard-Fejér type inequality for coordinated convex mappings which generalize the results (10), (12), (15), and (17).


## 2. Main Results

We will use the following lemma to prove our results.Lemma 10 (see [13]).
Letf
:
[
a
,
b
]
→
R be a convex function, 0
<
α
<
1, 0
<
β
<
1, A
=
α
a
+
(
1
-
α
)
b, u
0
=
(
b
-
a
)
min
⁡
{
α
/
(
1
-
β
)
,
(
1
-
α
)
/
β
}, and let h be defined by h
(
t
)
=
(
1
-
β
)
f
(
A
-
β
t
)
+
β
f
(
A
+
(
1
-
β
)
t
), t
∈
[
0
,
u
0
]. Then h is convex, increasing on [
0
,
u
0
], and, for all t
∈
[
0
,
u
0
],
(19)
f
(
α
a
+
(
1
-
α
)
b
)
≤
h
(
t
)
≤
α
f
(
a
)
+
(
1
-
α
)
f
(
b
)
.Lemma 11.
Letf
:
[
a
,
b
]
×
[
c
,
d
]
→
R be a coordinated convex function, 0
<
α
<
1, 0
<
β
<
1, 0
<
λ
<
1, 0
<
μ
<
1, A
=
α
a
+
(
1
-
α
)
b, B
=
λ
c
+
(
1
-
λ
)
d, u
0
=
(
b
-
a
)
min
⁡
{
α
/
1
-
β
,
(
1
-
α
)
/
β
}, v
0
=
(
d
-
c
)
min
⁡
{
λ
/
(
1
-
μ
)
,
(
1
-
λ
)
/
μ
} and let h be defined by
(20)
h
t
,
s
=
1
-
β
1
-
μ
f
A
-
β
t
,
B
-
μ
s
+
1
-
β
μ
f
A
-
β
t
,
B
+
1
-
μ
s
+
β
1
-
μ
f
A
+
1
-
β
t
,
B
-
μ
s
+
β
μ
f
A
+
1
-
β
t
,
B
+
1
-
μ
s
,
t
∈
[
0
,
u
0
], s
∈
[
0
,
v
0
]. Then h is coordinated convex, coordinated monotonic nondecreasing on [
0
,
u
0
]
×
[
0
,
v
0
], and, for all (
t
,
s
)
∈
[
0
,
u
0
]
×
[
0
,
v
0
],
(21)
f
(
α
a
+
(
1
-
α
)
b
,
λ
c
+
(
1
-
λ
)
d
)
≤
h
(
t
,
s
)
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
.Proof.
We note that iff is convex and g is linear, then the composition f
∘
g is convex. Also we note that a positive constant multiple of a convex function and a sum of two convex functions are convex. Hence, it is easily observed that h is coordinated convex, coordinated monotonic nondecreasing on [
0
,
u
0
]
×
[
0
,
v
0
], from the coordinated convexity of f and Lemma 10. Next,
(22)
h
t
,
s
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
,
B
-
μ
s
)
+
(
1
-
β
)
μ
f
(
A
-
β
t
,
B
+
(
1
-
μ
)
s
)
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
+
β
μ
f
(
A
+
(
1
-
β
)
t
,
B
+
(
1
-
μ
)
s
)
≥
f
(
1
-
β
)
(
A
-
β
t
)
+
β
(
A
+
(
1
-
β
)
t
)
,
dddd
(
1
-
μ
)
(
B
-
μ
s
)
+
μ
(
B
+
(
1
-
μ
)
s
)
=
f
(
A
,
B
)
=
f
(
α
a
+
(
1
-
α
)
b
,
λ
c
+
(
1
-
λ
)
d
)
.
Also,
(23)
h
(
t
,
s
)
=
(
1
-
β
)
(
1
-
μ
)
f
×
α
(
b
-
a
)
+
β
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
-
β
t
b
-
a
b
,
d
d
d
d
d
λ
(
d
-
c
)
+
μ
s
d
-
c
c
+
(
1
-
λ
)
(
d
-
c
)
-
μ
s
d
-
c
d
+
(
1
-
β
)
μ
f
×
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
α
(
b
-
a
)
+
β
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
-
β
t
b
-
a
b
,
d
d
d
d
d
λ
(
d
-
c
)
-
(
1
-
μ
)
s
d
-
c
c
d
d
d
d
d
+
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
+
β
(
1
-
μ
)
f
×
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
+
(
1
-
β
)
t
b
-
a
b
,
s
s
s
s
g
s
s
λ
(
d
-
c
)
+
μ
s
d
-
c
c
+
(
1
-
λ
)
(
d
-
c
)
-
μ
s
d
-
c
d
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
+
(
1
-
β
)
t
b
-
a
+
β
μ
f
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
a
dddddddd
+
(
1
-
α
)
(
b
-
a
)
+
(
1
-
β
)
t
b
-
a
b
,
d
d
d
d
d
d
d
d
λ
(
d
-
c
)
-
(
1
-
μ
)
s
d
-
c
c
d
d
d
d
d
d
d
d
+
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
=
(
1
-
β
)
(
1
-
μ
)
I
1
+
(
1
-
β
)
μ
I
2
+
β
(
1
-
μ
)
I
3
+
β
μ
I
4
.
From the coordinated convexity of f, we obtain
(24)
I
1
=
f
α
(
b
-
a
)
+
β
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
-
β
t
b
-
a
b
,
d
d
d
d
d
λ
(
d
-
c
)
+
μ
s
d
-
c
c
+
(
1
-
λ
)
(
d
-
c
)
-
μ
s
d
-
c
d
≤
α
b
-
a
+
β
t
b
-
a
λ
d
-
c
+
μ
s
d
-
c
f
a
,
c
+
α
b
-
a
+
β
t
b
-
a
1
-
λ
d
-
c
-
μ
s
d
-
c
f
a
,
d
+
1
-
α
b
-
a
-
β
t
b
-
a
λ
d
-
c
+
μ
s
d
-
c
f
b
,
c
+
1
-
α
b
-
a
-
β
t
b
-
a
1
-
λ
d
-
c
-
μ
s
d
-
c
×
f
b
,
d
,
I
2
=
f
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
α
(
b
-
a
)
+
β
t
b
-
a
a
+
(
1
-
α
)
(
b
-
a
)
-
β
t
b
-
a
b
,
f
f
f
f
f
λ
(
d
-
c
)
-
(
1
-
μ
)
s
d
-
c
c

+

(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
≤
α
b
-
a
+
β
t
b
-
a
λ
d
-
c
-
1
-
μ
s
d
-
c
f
a
,
c

+

α
b
-
a
+
β
t
b
-
a

1
-
λ
d
-
c
+
1
-
μ
s
d
-
c
f
a
,
d

+

1
-
α
b
-
a
-
β
t
b
-
a
λ
d
-
c
-
1
-
μ
s
d
-
c
f
b
,
c
+
1
-
α
b
-
a
-
β
t
b
-
a
1
-
λ
d
-
c
+
1
-
μ
s
d
-
c
×
f
b
,
d
,
I
3
=
f
(
1
-
λ
)
(
d
-
c
)
-
μ
s
d
-
c
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
a
ddddd
+
(
1
-
α
)
(
b
-
a
)
+
(
1
-
β
)
t
b
-
a
b
,
d
d
d
d
d
λ
(
d
-
c
)
+
μ
s
d
-
c
c
+
(
1
-
λ
)
(
d
-
c
)
-
μ
s
d
-
c
d
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
≤
α
b
-
a
-
1
-
β
t
b
-
a
λ
d
-
c
+
μ
s
d
-
c
f
a
,
c

+

α
b
-
a
-
1
-
β
t
b
-
a

1
-
λ
d
-
c
-
μ
s
d
-
c
f
a
,
d

+

1
-
α
b
-
a
+
1
-
β
t
b
-
a
λ
d
-
c
+
μ
s
d
-
c
f
b
,
c
+
1
-
α
b
-
a
+
1
-
β
t
b
-
a
1
-
λ
d
-
c
-
μ
s
d
-
c
×
f
(
b
,
d
)
,
I
4
=
f
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a
a
d
d
d
d
d
+
1
-
α
b
-
a
+
1
-
β
t
b
-
a
b
,
s
s
s
s
s
s
d
s
λ
(
d
-
c
)
-
(
1
-
μ
)
s
d
-
c
c
d
d
d
d
d
+
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
d
-
c
d
α
(
b
-
a
)
-
(
1
-
β
)
t
b
-
a

≤

α
b
-
a
-
1
-
β
t
b
-
a
λ
d
-
c
-
1
-
μ
s
d
-
c
f
a
,
c
+
α
b
-
a
-
1
-
β
t
b
-
a
×
1
-
λ
d
-
c
+
1
-
μ
s
d
-
c
f
(
a
,
d
)

+

1
-
α
b
-
a
+
1
-
β
t
b
-
a
λ
d
-
c
-
1
-
μ
s
d
-
c
×
f
(
b
,
c
)
+
1
-
α
b
-
a
+
1
-
β
t
b
-
a
×
1
-
λ
d
-
c
+
1
-
μ
s
d
-
c
f
(
b
,
d
)
.
We note that 0
<
α
≤
(
α
(
b
-
a
)
+
β
t
)
/
(
b
-
a
)
≤
1, 0
≤
(
(
1
-
α
)
(
b
-
a
)
-
β
t
)
/
(
b
-
a
)
≤
1, 0
≤
(
α
(
b
-
a
)
-
(
1
-
β
)
t
)
/
(
b
-
a
)
≤
α
≤
1, 0
<
1
-
α
≤
(
(
1
-
α
)
(
b
-
a
)
+
(
1
-
β
)
t
)
/
(
b
-
a
)
≤
1, 0
<
λ
≤
(
λ
(
d
-
c
)
+
μ
s
)
/
(
d
-
c
)
≤
1, 0
≤
(
(
1
-
λ
)
(
d
-
c
)
-
μ
s
)
/
(
d
-
c
)
≤
1, 0
≤
(
λ
(
d
-
c
)
-
(
1
-
μ
)
s
)
/
(
d
-
c
)
≤
λ
≤
1, 0
<
1
-
λ
≤
(
(
1
-
λ
)
(
d
-
c
)
+
(
1
-
μ
)
s
)
/
(
d
-
c
)
≤
1. So
(25)
h
t
,
s
=
(
1
-
β
)
(
1
-
μ
)
I
1
+
(
1
-
β
)
μ
I
2
+
β
(
1
-
μ
)
I
3
+
β
μ
I
4
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
,
which completes the proof.Theorem 12.
Letf, A, B, α, β, λ, μ, u
0, and v
0 be defined as in Lemma 11, and let g
:
[
a
,
b
]
×
[
c
,
d
]
→
R be nonnegative and integrable and
(26)
g
A
-
β
u
,
s
=
g
A
+
1
-
β
u
,
s
,
u
,
s
∈
0
,
u
0
×
c
,
d
,
g
(
t
,
B
-
μ
s
)
=
g
(
t
,
B
+
(
1
-
μ
)
s
)
,
(
t
,
s
)
∈
[
a
,
b
]
×
[
0
,
v
0
]
.
Then
(27)
f
α
a
+
1
-
α
b
,
λ
c
+
1
-
λ
d
×
∫
A
-
β
u
A
+
1
-
β
u
‍
∫
B
-
μ
v
B
+
1
-
μ
v
‍
g
t
,
s
d
t
d
s
≤
(
1
-
β
)
(
1
-
μ
)
β
μ
∫
A
-
β
u
A
‍
∫
B
-
μ
v
B
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
(
1
-
β
)
μ
β
(
1
-
μ
)
∫
A
-
β
u
A
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
β
(
1
-
μ
)
(
1
-
β
)
μ
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
β
μ
(
1
-
β
)
(
1
-
μ
)
×
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
s
s
s
s
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
×
∫
A
-
β
u
A
+
1
-
β
u
‍
∫
B
-
μ
v
B
+
1
-
μ
v
‍
g
t
,
s
d
t
d
s
.Proof.
For every(
u
,
v
)
∈
[
0
,
u
0
]
×
[
0
,
v
0
], we have the identity
(28)
∫
A
-
β
u
A
+
1
-
β
u
‍
∫
B
-
μ
v
B
+
1
-
μ
v
‍
g
t
,
s
d
t
d
s
=
∫
B
-
μ
v
B
+
1
-
μ
v
‍
∫
A
-
β
u
A
+
1
-
β
u
‍
g
t
,
s
d
t
d
s
=
∫
B
-
μ
v
B
+
1
-
μ
v
‍
∫
A
-
β
u
A
‍
g
t
,
s
d
t
+
∫
A
A
+
1
-
β
u
‍
g
t
,
s
d
t
d
s
=
∫
B
-
μ
v
B
+
(
1
-
μ
)
v
‍
β
∫
0
u
‍
g
(
A
-
β
t
,
s
)
d
t
d
d
d
d
d
d
d
d
d
d
+
(
1
-
β
)
∫
0
u
‍
g
(
A
-
β
t
,
s
)
d
t
d
s
=
∫
B
-
μ
v
B
+
1
-
μ
v
‍
∫
0
u
‍
g
A
-
β
t
,
s
d
t
d
s
=
∫
0
u
‍
∫
B
-
μ
v
B
+
1
-
μ
v
‍
g
A
-
β
t
,
s
d
s
d
t
=
∫
0
u
‍
∫
0
v
‍
g
A
-
β
t
,
B
-
μ
s
d
t
d
s
.
Since g is nonnegative, multiplying (21) by g
(
A
-
β
t
,
B
-
μ
s
), integrating the resulting inequalities over [
0
,
u
]
×
[
0
,
v
], and using (26), we have
(29)
g
A
-
β
t
,
B
-
μ
s
=
g
A
-
β
t
,
B
+
1
-
μ
s
ssssssssssssdssssss
=
g
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
ssssssssssdssssssss
=
g
A
+
1
-
β
t
,
B
+
1
-
μ
s
,
f
(
α
a
+
(
1
-
α
)
b
,
λ
c
+
(
1
-
λ
)
d
)
×
∫
0
u
‍
∫
0
v
‍
g
A
-
β
t
,
B
-
μ
s
d
t
d
s
≤
(
1
-
β
)
(
1
-
μ
)
×
∫
0
u
‍
∫
0
v
‍
f
(
A
-
β
t
,
B
-
μ
s
)
g
(
A
-
β
t
,
B
-
μ
s
)
d
t
d
s
+
(
1
-
β
)
μ
∫
0
u
‍
∫
0
v
‍
f
(
A
-
β
t
,
B
+
(
1
-
μ
)
s
)
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
×
g
(
A
-
β
t
,
B
+
(
1
-
μ
)
s
)
d
t
d
s
+
β
(
1
-
μ
)
∫
0
u
‍
∫
0
v
‍
f
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
×
g
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
d
t
d
s
+
β
μ
∫
0
u
‍
∫
0
v
‍
f
(
A
+
(
1
-
β
)
t
,
B
+
(
1
-
μ
)
s
)
d
d
d
d
d
d
d
d
d
d
d
×
g
(
A
+
(
1
-
β
)
t
,
B
+
(
1
-
μ
)
s
)
d
t
d
s
=
(
1
-
β
)
(
1
-
μ
)
β
μ
∫
A
-
β
u
A
‍
∫
B
-
μ
v
B
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
(
1
-
β
)
μ
β
(
1
-
μ
)
∫
A
-
β
u
A
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
β
(
1
-
μ
)
(
1
-
β
)
μ
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
+
β
μ
(
1
-
β
)
(
1
-
μ
)
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
t
,
s
)
g
(
t
,
s
)
d
t
d
s
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
sss
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
×
∫
0
u
‍
∫
0
v
‍
g
(
A
-
β
t
,
B
-
μ
s
)
d
t
d
s
.
Thus, inequalities (27) follow by using the identity (28).Remark 13.
If we chooseα
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c in Theorem 12, then inequalities (27) reduce to inequalities (14).Remark 14.
If we chooseα
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c, and g
(
x
,
y
)
≡
1 in Theorem 12, then inequalities (27) reduce to inequalities (9).Theorem 15.
Letf, A, B, u
0, and v
0 be defined as in Lemma 11, 0
<
α
<
1, 0
<
β
<
1, α
+
β
≤
1, 0
<
λ
<
1, 0
<
μ
<
1, λ
+
μ
≤
1, and let H be defined on [
0,1
]
2 by(30)
H
t
,
s
=
(
1
-
β
)
(
1
-
μ
)
α
λ
(
b
-
a
)
(
d
-
c
)
×
∫
0
α
(
b
-
a
)
/
(
1
-
β
)
‍
∫
0
λ
(
d
-
c
)
/
(
1
-
μ
)
‍
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
x
,
B
-
μ
s
y
)
+
(
1
-
β
)
μ
f
(
A
-
β
t
x
,
B
+
(
1
-
μ
)
s
y
)
ssssssssssssssssssssssssssssssssss
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
t
x
,
B
-
μ
s
y
)
+
β
μ
f
A
+
1
-
β
t
x
,
B
+
1
-
μ
s
y
d
y
d
x
.Then, H is coordinated convex and coordinated monotonic nondecreasing on [
0,1
]
2, and(31)
f
(
α
a
+
(
1
-
α
)
b
,
λ
c
+
(
1
-
λ
)
d
)
=
H
(
0,0
)
≤
H
(
t
,
s
)
≤
H
(
1,1
)
=
(
1
-
β
)
(
1
-
μ
)
α
λ
(
b
-
a
)
(
d
-
c
)
×
∫
0
α
(
b
-
a
)
/
(
1
-
β
)
‍
∫
0
λ
(
d
-
c
)
/
(
1
-
μ
)
‍
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
x
,
B
-
μ
y
)
+
(
1
-
β
)
μ
f
(
A
-
β
x
,
B
+
(
1
-
μ
)
y
)
ddd
d
d
d
d
d
d
d
ddddddddddddd
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
x
,
B
-
μ
y
)
+
β
μ
f
A
+
1
-
β
x
,
B
+
1
-
μ
y
d
y
d
x
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
.Proof.
ThatH is coordinated convex follows immediately from the coordinated convexity of f. Next, the conditions α
+
β
≤
1 and λ
+
μ
≤
1 imply that u
0
=
α
(
b
-
a
)
/
(
1
-
β
) and v
0
=
λ
(
d
-
c
)
/
(
1
-
μ
), respectively. It follows from Lemma 11 that h
(
t
,
s
)
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
,
B
-
μ
s
)
+ (
1
-
β
)
μ
f
(
A
-
β
t
,
B
+
(
1
-
μ
)
s
)
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
+
β
μ
f
(
A
+
(
1
-
β
)
t
,
B
+
(
1
-
μ
)
s
) is coordinated monotonic nondecreasing on [
0
,
u
0
]
×
[
0
,
v
0
] and hence H
(
t
,
s
) is coordinated monotonic nondecreasing on [
0,1
]
2. Finally, the last inequality of (31) follows from (21), and the proof is completed.Similarly, we have the following theorem.Theorem 16.
Letf, A, B, u
0, and v
0 be defined as in Lemma 11,   0
<
α
<
1, 0
<
β
<
1, α
+
β
≤
1, 0
<
λ
<
1, 0
<
μ
<
1, λ
+
μ
≤
1, and let G be defined on [
0,1
]
2 by(32)
G
t
,
s
=
(
1
-
β
)
(
1
-
μ
)
α
λ
(
b
-
a
)
(
d
-
c
)
×
∫
0
α
b
-
a
/
1
-
β
‍
∫
0
λ
d
-
c
/
1
-
μ
‍
B
+
1
-
μ
λ
d
-
c
1
-
μ
-
y
1
-
s
1
-
β
1
-
μ
f
A
-
β
α
b
-
a
1
-
β
-
x
1
-
t
,
B
-
μ
λ
d
-
c
1
-
μ
-
y
1
-
s
d
d
d
d
d
d
d
d
d
d
d
d
s
s
s
d
d
s
d
d
d
d
d
+
1
-
β
μ
f
A
-
β
α
b
-
a
1
-
β
-
x
1
-
t
,
B
+
1
-
μ
λ
d
-
c
1
-
μ
-
y
1
-
s
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
+
β
(
1
-
μ
)
f
A
+
1
-
β
α
b
-
a
1
-
β
-
x
1
-
t
,
B
-
μ
λ
d
-
c
1
-
μ
-
y
1
-
s
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
+
β
μ
f
A
+
1
-
β
α
b
-
a
1
-
β
-
x
1
-
t
,
B
+
1
-
μ
λ
d
-
c
1
-
μ
-
y
1
-
s
d
y
d
x
.Then, G is coordinated convex and coordinated monotonic nondecreasing on [
0,1
]
2, and
(33)
1
-
β
2
1
-
μ
2
β
μ
α
λ
b
-
a
d
-
c
×
∫
b
-
α
b
-
a
/
1
-
β
A
‍
∫
d
-
λ
d
-
c
/
1
-
μ
B
‍
f
x
,
y
d
y
d
x
+
1
-
β
2
μ
β
α
λ
(
b
-
a
)
(
d
-
c
)
∫
b
-
α
(
b
-
a
)
/
(
1
-
β
)
A
‍
∫
B
d
‍
f
(
x
,
y
)
d
y
d
x
+
β
1
-
μ
2
μ
α
λ
(
b
-
a
)
(
d
-
c
)
∫
A
b
‍
∫
d
-
λ
(
d
-
c
)
/
1
-
μ
B
‍
f
(
x
,
y
)
d
y
d
x
+
β
μ
α
λ
(
b
-
a
)
(
d
-
c
)
∫
A
b
‍
∫
B
d
‍
f
(
x
,
y
)
d
y
d
x
=
G
(
0,0
)
≤
G
(
t
,
s
)
≤
G
(
1,1
)
=
1
-
β
1
-
μ
f
b
-
α
b
-
a
1
-
β
,
d
-
λ
d
-
c
1
-
μ
+
1
-
β
μ
f
b
-
α
b
-
a
1
-
β
,
d
+
β
1
-
μ
f
b
,
d
-
λ
d
-
c
1
-
μ
+
β
μ
f
(
b
,
d
)
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
1
-
λ
f
a
,
d
+
1
-
α
1
-
λ
f
b
,
d
.Proof.
The proof is similar to that of Theorem15, so we omit the details.Remark 17.
Identity (10) is a special case of (30) if we choose α
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c.Remark 18.
Identity (12) is a special case of (32) if we choose α
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c.Theorem 19.
Letf, A, B, u
0, and v
0 be defined as in Lemma 11, 0
<
α
<
1, 0
<
β
<
1, α
+
β
≤
1, 0
<
λ
<
1, 0
<
μ
<
1, λ
+
μ
≤
1, and let g be defined as in Theorem 12. Let G be defined on [
0,1
]
×
[
0,1
] by
(34)
P
(
t
,
s
)
=
∫
0
u
‍
∫
0
v
‍
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
x
,
B
-
μ
s
y
)
s
s
s
s
s
s
s
s
s
s
s
×
g
(
A
-
β
x
,
B
-
μ
y
)
s
s
s
s
s
s
s
s
s
s
s
+
(
1
-
β
)
μ
f
(
A
-
β
t
x
,
B
+
(
1
-
μ
)
s
y
)
s
s
s
s
s
s
s
s
s
s
s
×
g
(
A
-
β
x
,
B
+
(
1
-
μ
)
y
)
s
s
s
s
s
s
s
s
s
s
s
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
t
x
,
B
-
μ
s
y
)
s
s
s
s
s
s
s
s
s
s
s
×
g
(
A
+
(
1
-
β
)
x
,
B
-
μ
y
)
s
s
s
s
s
s
s
s
s
s
s
+
β
μ
f
(
A
+
(
1
-
β
)
t
x
,
B
+
(
1
-
μ
)
s
y
)
s
s
s
s
s
s
s
s
s
s
s
×
g
A
+
1
-
β
x
,
B
+
1
-
μ
y
d
x
d
y
,
for some u
∈
[
0
,
u
0
], v
∈
[
0
,
v
0
]. Then P
(
t
,
s
) is coordinated convex and coordinated monotonic nondecreasing on [
0,1
]
2, and
(35)
f
α
a
+
1
-
α
b
,
λ
c
+
1
-
λ
d
×
∫
A
-
β
u
A
+
1
-
β
u
‍
∫
B
-
μ
v
B
+
1
-
μ
v
‍
g
x
,
y
d
y
d
x
=
P
(
0,0
)
≤
P
(
t
,
s
)
≤
P
(
1,1
)
=
(
1
-
β
)
(
1
-
μ
)
×
∫
0
u
‍
∫
0
v
‍
f
A
-
β
t
,
B
-
μ
s
g
A
-
β
t
,
B
-
μ
s
d
t
d
s
+
(
1
-
β
)
μ
×
∫
0
u
‍
∫
0
v
‍
f
A
-
β
t
,
B
+
1
-
μ
s
d
d
d
d
d
d
d
×
g
A
-
β
t
,
B
+
1
-
μ
s
d
t
d
s
+
β
(
1
-
μ
)
×
∫
0
u
‍
∫
0
v
‍
f
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
f
f
f
f
f
g
g
f
f
f
×
g
(
A
+
(
1
-
β
)
t
,
B
-
μ
s
)
d
t
d
s
+
β
μ
∫
0
u
‍
∫
0
v
‍
f
A
+
1
-
β
t
,
B
+
1
-
μ
s
ddddddddddd
×
g
A
+
1
-
β
t
,
B
+
1
-
μ
s
d
t
d
s
=
(
1
-
β
)
(
1
-
μ
)
β
μ
∫
A
-
β
u
A
‍
∫
B
-
μ
v
B
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
+
(
1
-
β
)
μ
β
(
1
-
μ
)
∫
A
-
β
u
A
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
+
β
(
1
-
μ
)
(
1
-
β
)
μ
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
+
β
μ
(
1
-
β
)
(
1
-
μ
)
×
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
.Proof.
Sincef is coordinated convex and g is nonnegative, we see that P
(
t
,
s
) is coordinated convex on [
0,1
]
2. Next, for each (
x
,
y
)
∈
[
0
,
u
]
×
[
0
,
v
], where u
∈
[
0
,
u
0
], v
∈
[
0
,
v
0
], it follows from Lemma 11 that h
(
t
x
,
t
y
)
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
x
,
B
-
μ
s
y
)
+
(
1
-
β
)
μ
f
(
A
-
β
t
x
,
B
+
(
1
-
μ
)
s
y
)
+
β
(
1
-
μ
)
f
(
A
+
(
1
- β
)
t
x
,
B
-
μ
s
y
)
+
β
μ
f
(
A
+
(
1
-
β
)
t
x
,
B
+
(
1
-
μ
)
s
y
) is coordinated monotonic nondecreasing on [
0,1
]
2. Using (26) we see that P
(
t
,
s
) is coordinated monotonic nondecreasing on [
0,1
]
2, which completes the proof.Theorem 20.
Letf, A, B, u
0, and v
0 be defined as in Lemma 11, 0
<
α
<
1, 0
<
β
<
1, α
+
β
≤
1, 0
<
λ
<
1, 0
<
μ
<
1, λ
+
μ
≤
1, and let g be defined as in Theorem 12. Let Q
(
t
,
s
) be defined on [
0,1
]
2 by
(36)
Q
(
t
,
s
)
=
∫
0
u
‍
∫
0
v
‍
(
1
-
β
)
(
1
-
μ
)
s
s
s
s
s
s
s
s
s
s
s
×
f
A
-
β
u
+
β
x
1
-
t
,
B
-
μ
v
+
μ
1
-
s
y
d
d
d
d
d
d
i
d
×
g
A
-
β
u
-
x
,
B
-
μ
v
-
y
ssssssssssss
+
(
1
-
β
)
μ
f
A
-
β
u
+
β
x
(
1
-
t
)
,
B
+
(
1
-
μ
)
v
ddddddddsddddddddddd
-
(
1
-
μ
)
(
1
-
s
)
y
ssssssssssss
×
g
A
-
β
u
-
x
,
B
+
1
-
μ
v
-
y
ssssssssssss
+
β
(
1
-
μ
)
f
A
+
(
1
-
β
)
u
-
(
1
-
β
)
x
(
1
-
t
)
,
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
B
-
μ
v
+
μ
(
1
-
s
)
y
d
d
d
d
s
d
d
d
×
g
A
+
1
-
β
u
-
x
,
B
-
μ
v
-
y
d
d
d
d
s
d
d
d
+
β
μ
f
A
+
(
1
-
β
)
u
-
(
1
-
β
)
x
(
1
-
t
)
,
d
d
d
d
d
s
d
d
d
d
d
d
d
d
B
+
1
-
μ
v
-
1
-
μ
y
1
-
s
f
f
f
f
f
s
f
f
f
f
×
g
A
+
(
1
-
β
)
(
u
-
x
)
,
d
d
d
d
d
d
d
d
d
d
d
d
B
+
1
-
μ
v
-
y
d
y
d
x
,
for some u
∈
[
0
,
u
0
], v
∈
[
0
,
v
0
]. Then Q
(
t
,
s
) is coordinated convex and coordinated monotonic nondecreasing on [
0,1
]
2, and
(37)
1
-
β
1
-
μ
β
μ
∫
A
-
β
u
A
‍
∫
B
-
μ
v
B
‍
f
x
,
y
g
x
,
y
d
x
d
y
+
(
1
-
β
)
μ
β
(
1
-
μ
)
∫
A
-
β
u
A
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
+
β
(
1
-
μ
)
(
1
-
β
)
μ
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
+
β
μ
(
1
-
β
)
(
1
-
μ
)
×
∫
A
A
+
(
1
-
β
)
u
‍
∫
B
B
+
(
1
-
μ
)
v
‍
f
(
x
,
y
)
g
(
x
,
y
)
d
x
d
y
=
Q
(
0,0
)
≤
Q
(
t
,
s
)
≤
Q
(
1,1
)
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
u
,
B
-
μ
v
)
ddd
+
(
1
-
β
)
μ
f
(
A
-
β
u
,
B
+
(
1
-
μ
)
v
)
ddd
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
u
,
B
-
μ
v
)
ddd
+
β
μ
f
(
A
+
(
1
-
β
)
u
,
B
+
(
1
-
μ
)
v
)
×
∫
A
-
β
u
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
+
(
1
-
μ
)
v
‍
g
(
x
,
y
)
d
x
d
y
≤
α
λ
f
(
a
,
c
)
+
(
1
-
α
)
λ
f
(
b
,
c
)
+
α
(
1
-
λ
)
f
(
a
,
d
)
f
f
f
+
(
1
-
α
)
(
1
-
λ
)
f
(
b
,
d
)
×
∫
A
-
β
u
A
+
(
1
-
β
)
u
‍
∫
B
-
μ
v
B
+
(
1
-
μ
)
v
‍
g
(
x
,
y
)
d
x
d
y
.Proof.
ThatH is coordinated convex follows immediately from the coordinated convexity of f. Next, for each (
x
,
y
)
∈
[
0
,
u
]
×
[
0
,
v
], where u
∈
[
0
,
u
0
], v
∈
[
0
,
v
0
], it follows from Lemma 11 that h
(
t
,
s
)
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
t
,
B
-
μ
s
)
+
(
1
-
β
)
μ
f
(
A
-
β
t
,
B
+
(
1
-
μ
)
s
)
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
t,B
-
μ
s
)
+
β
μ
f
(
A
+
(
1
-
β
)
t
,
B
+
(
1
-
μ
)
s
) is coordinated monotonic nondecreasing on [
u
,
v
]
∈
[
0
,
u
0
]
×
[
0
,
v
0
]. Since k
1
(
t
)
=
u
-
(
1
-
t
)
x and k
2
(
s
)
=
v
-
(
1
-
s
)
y are increasing on [
0,1
], respectively, we get that h
(
k
1
(
t
)
,
k
2
(
s
)
)
=
(
1
-
β
)
(
1
-
μ
)
f
(
A
-
β
u
+
β
x
(
1
-
t
)
,
B
-
μ
v
+
μ
(
1
-
s
)
y
)
+
(
1
-
β
)
μ
f
(
A
-
β
u
+
β
x
(
1
-
t
)
,
B
+
(
1
-
μ
)
v
-
(
1
-
μ
)
(
1
-
s
)
y
)
+
β
(
1
-
μ
)
f
(
A
+
(
1
-
β
)
u
-
(
1
-
β
)
x
(
1
-
t
)
,
B
-
μ
v
+
μ
(
1
-
s
)
y
)
+
β
μ
f
(
A
+
(
1
-
β
)
u
-
(
1
-
β
)
x
(
1
-
t
)
,
B
+
(
1
-
μ
)
v
-
(
1
-
μ
)
(
1
-
s
)
y
) is coordinated monotonic nondecreasing on [
0,1
]
2. Since g is nonnegative and satisfies (26), it follows that Q
(
t
,
s
) is coordinated monotonic nondecreasing on [
0,1
]
2. Finally, the last inequality of (37) follows from (21), and the proof is completed.Remark 21.
Identity (15) is a special case of (34) if we choose α
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c.Remark 22.
Identity (17) is a special case of (36) if we choose α
=
β
=
λ
=
μ
=
1
/
2, u
=
u
0
=
b
-
a, v
=
v
0
=
d
-
c.



---

*Source: 796132-2014-11-17.xml*