Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 8 (2015), 740–749
Research Article
Some new Hermite–Hadamard type inequalities for
geometrically quasi-convex functions on co-ordinates
Xu-Yang Guoa , Feng Qib,∗, Bo-Yan Xia
a
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China
b
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China
Abstract
In the paper, the authors introduce a new concept “geometrically quasi-convex function on co-ordinates”
and establish some new Hermite–Hadamard type inequalities for geometrically quasi-convex functions on
c
the co-ordinates. 2015
All rights reserved.
Keywords: Geometrically quasi-convex function, Hermite–Hadamard type integral inequality, Hölder
inequality.
2010 MSC: 26A51, 26D15.
1. Introduction
The following definitions are well known in the literature.
Definition 1.1. A function f : I ⊆ R = (−∞, ∞) → R is said to be convex if
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y)
is valid for all x, y ∈ I and λ ∈ [0, 1].
Definition 1.2 ([6]). A function f : I ⊆ R → R is said to be quasi-convex if
f (λx + (1 − λ)y) ≤ max{f (x), f (y)}
holds for all x, y ∈ I and λ ∈ [0, 1].
∗
Corresponding author
Email addresses: guoxuyang1991@qq.com (Xu-Yang Guo), qifeng618@gmail.com, qifeng618@hotmail.com (Feng Qi),
baoyintu78@qq.com (Bo-Yan Xi)
Received 2015-2-28
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
741
Definition 1.3 ([4, 5]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R is said to be convex on the co-ordinates
on ∆ with a < b and c < d if the partial mappings
fy : [a, b] → R,
fy (u) = fy (u, y)
and
fx : [c, d] → R,
fx (v) = fx (x, v)
are convex for all x ∈ (a, b) and y ∈ (c, d).
A formal definition for co-ordinated convex functions may be restated as follows.
Definition 1.4 ([4, 5]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R is said to be convex on the co-ordinates
on ∆ with a < b and c < d if
f (tx + (1 − t)z, λy + (1 − λ)w) ≤ tλf (x, y) + t(1 − λ)f (x, w) + (1 − t)λf (z, y) + (1 − t)(1 − λ)f (z, w)
holds for all t, λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆.
Definition 1.5 ([9]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R is said to be a quasi-convex on the
co-ordinates on ∆ with a < b and c < d if
f (λx + (1 − λ)z, λy + (1 − λ)w) ≤ max{f (x, y), f (z, w)}
holds for all λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆.
A formal definition for co-ordinated quasi-convex functions may be stated as follows.
Definition 1.6 ([10]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R is said to be quasi-convex on the
co-ordinates on ∆ with a < b and c < d if
f (λx + (1 − λ)z, λy + (1 − λ)w) ≤ max{f (x, y), f (x, w), f (z, y), f (z, w)}
holds for all λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆.
For convex functions on the co-ordinates, there exist the following conclusions.
Theorem 1.7 ([4, 5, Theorem 2.2]). Let f : ∆ = [a, b] × [c, d] ⊆ R2 → R be convex on the co-ordinates on
∆ with a < b and c < d. Then
Z b Z d a+b c+d
1
1
c+d
1
a+b
f
,
≤
f x,
dx +
f
,y dy
2
2
2 b−a a
2
d−c c
2
Z dZ b
1
≤
f (x, y) d x d y
(b − a)(d − c) c a
Z b
Z d
Z b
Z d
1
1
1
≤
f (x, c) d x +
f (x, d) d x +
f (a, y) d y +
f (b, y) d y
4 b−a a
d−c c
a
c
1
≤ [f (a, c) + f (b, c) + f (a, d) + f (b, d)].
4
Theorem 1.8 ([10, Theorem 2.1]). Let f : ∆ = [a, b] × [c, d] ⊆ R2 → R be a partial differentiable function
∂2f is quasi-convex on the co-ordinates on ∆, then
on ∆ with a < b and c < d. If ∂x∂y
Z dZ b
f (a, c) + f (a, d) + f (b, c) + f (b, d)
1
+
f (x, y) d x d y − A
4
(b − a)(d − c) c a
2
2
2
∂ f (a, c) ∂ f (a, d) ∂ f (b, c) ∂ 2 f (b, d) (b − a)(d − c)
,
≤
,
,
,
max 16
∂x∂y ∂x∂y ∂x∂y ∂x∂y where
A=
1
2(b − a)
Z
b
[f (x, c) + f (x, d)] d x +
a
1
2(d − c)
Z
d
[f (a, y) + f (b, y)] d y.
c
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
742
For more information on this topic, please refer to the papers [1, 2, 3, 7, 8, 11, 12, 13, 14, 15] and related
references therein.
In this paper, we will introduce a new concept “geometrically quasi-convex function on co-ordinates”
and establish some new Hermite–Hadamard type inequalities for geometrically quasi-convex functions on
the co-ordinates.
2. Definition and Lemmas
Now we introduce the definition of the geometrically quasi-convex functions.
Definition 2.1. Let R+ = (0, ∞). A function f : ∆ = [a, b] × [c, d] ⊆ R2+ → R is said to be geometrically
quasi-convex on the co-ordinates on ∆ with a < b and c < d if
f xt z 1−t , y λ w1−λ ≤ max{f (x, y), f (x, w), f (z, y), f (z, w)}
holds for all t, λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆.
Remark 2.2. If f : ∆ ⊆ R2+ → R is increasing and convex on the co-ordinates on ∆, then it is geometrically
quasi-convex on the co-ordinates on ∆. If f : ∆ ⊆ R2+ → R is decreasing and geometrically quasi-convex on
the co-ordinates on ∆, then it is quasi-convex on the co-ordinates on ∆.
Proof. By Definitions 1.4, 1.6, and 2.1, we have
f xt z 1−t , y λ w1−λ ≤ f (tx + (1 − t)z, λy + (1 − λ)w)
≤ tλf (x, y) + t(1 − λ)f (x, w) + (1 − t)λf (z, y) + (1 − t)(1 − λ)f (z, w)
≤ max{f (x, y), f (x, w), f (z, y), f (z, w)}
and
f (λx + (1 − λ)z, λy + (1 − λ)w) ≤ f xλ z 1−λ , y λ w1−λ ≤ max{f (x, y), f (x, w), f (z, y), f (z, w)}.
This completes the required proof.
In order to prove our main results, we need the following integral identity.
Lemma 2.3. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R have partial derivatives of the second order with a < b
∂2f
and c < d. If ∂x∂y
∈ L1 (∆), then
√ Z b
√
√ f x, cd
16
1
f ab , cd −
dx
S(f ) ,
(ln b − ln a)(ln d − ln c)
ln b − ln a a
x
Z d √
Z dZ b
f ab , y
1
1
f (x, y)
−
dy +
dxdy
ln d − ln c c
y
(ln b − ln a)(ln d − ln c) c a
xy
Z 1Z 1
1−t 1+t 1−λ 1+λ
∂ 2 1−t 1+t 1−λ 1+λ =
(1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
Z 1Z 1
1−t 1+t 1+λ 1−λ
∂ 2 1−t 1+t 1+λ 1−λ −
(1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
Z 1Z 1
1+t 1−t 1−λ 1+λ
∂ 2 1+t 1−t 1−λ 1+λ −
(1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
Z 1Z 1
1+t 1−t 1+λ 1−λ
∂ 2 1+t 1−t 1+λ 1−λ +
(1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 , c 2 d 2 d t d λ.
∂x∂y
0
0
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
Proof. Integrating by parts gives
Z 1Z 1
1−t 1+t 1−λ 1+λ
∂ 2 1−t 1+t 1−λ 1+λ (1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
Z 1
1−λ 1+λ
∂ 1−t 1+t 1−λ 1+λ 1
2
(1 − λ)c 2 d 2 (1 − t) f a 2 b 2 , c 2 d 2 =
ln b − ln a 0
∂y
0
Z 1
1−λ 1+λ
1−t 1+t
∂
+
f a 2 b 2 ,c 2 d 2 dt dλ
0 ∂y
Z 1
√
1−λ 1+λ ∂
1−λ 1+λ
2
=
(1 − λ)c 2 d 2
f
ab , c 2 d 2 d λ
ln a − ln b 0
∂y
Z 1Z 1
1−λ 1+λ ∂
1−t 1+t
1−λ 1+λ
2
2
2
2
2
2
−
(1 − λ)c
d
f a b ,c
d
dtdλ
∂y
0
0
√
√ Z 1 √
1−λ 1+λ
4
f ab , cd −
f
ab , c 2 d 2 d λ
=
(ln b − ln a)(ln d − ln c)
0
Z 1Z 1 Z 1 1−t 1+t
1−λ 1+λ
1−t 1+t √
f a 2 b 2 ,c 2 d 2 dtdλ .
−
f a 2 b 2 , cd d t +
0
0
1−t
2
1+t
2
0
1−λ
2
1+λ
Choosing in the above identity x = a b
and y = c
d 2 for t, λ ∈ [0, 1] yields
Z 1Z 1
1−t 1+t 1−λ 1+λ
∂ 2 1−t 1+t 1−λ 1+λ (1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
√ Z b
√
√ f x, cd
4
2
=
f ab , cd −
dx
(ln b − ln a)(ln d − ln c)
ln b − ln a √ab
x
√
Z d
Z d Z b
f ab , y
f (x, y)
2
4
−
dy +
dxdy .
ln d − ln c √cd
y
(ln b − ln a)(ln d − ln c) √cd √ab xy
Similarly, we obtain
Z 1Z 1
1−t 1+t 1+λ 1−λ
∂ 2 1−t 1+t 1+λ 1−λ (1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
√ Z b
√
√ f x, cd
4
2
=−
f ab , cd −
dx
(ln b − ln a)(ln d − ln c)
ln b − ln a √ab
x
√
√
√
Z cd
Z cd Z b
f ab , y
f (x, y)
2
4
−
dy +
dxdy ,
√
ln d − ln c c
y
(ln b − ln a)(ln d − ln c) c
xy
ab
Z 1Z 1
1+t 1−t 1−λ 1+λ
∂ 2 1+t 1−t 1−λ 1+λ (1 − t)(1 − λ)a 2 b 2 c 2 d 2
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
0
√ Z √ab
√
√ f x, cd
4
2
=−
f ab , cd −
dx
(ln b − ln a)(ln d − ln c)
ln b − ln a a
x
√
Z d
Z d Z √ab
f ab , y
2
4
f (x, y)
−
dy +
dxdy ,
ln d − ln c √cd
y
(ln b − ln a)(ln d − ln c) √cd a
xy
and
Z
0
1Z 1
∂ 2 1+t 1−t 1+λ 1−λ (1 − t)(1 − λ)a b c
d
f a 2 b 2 ,c 2 d 2 dtdλ
∂x∂y
0
√ Z √ab
√
√ f x, cd
4
2
=
f ab , cd −
dx
(ln b − ln a)(ln d − ln c)
ln b − ln a a
x
√
Z √cd
Z √cd Z √ab
f ab , y
2
f (x, y)
4
−
dy +
dxdy .
ln d − ln c c
y
(ln b − ln a)(ln d − ln c) c
xy
a
1+t
2
1−t
2
1+λ
2
1−λ
2
743
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
744
This completes the proof of Lemma 2.3.
Lemma 2.4. Let u, v > 0, h ∈ R, and h 6= 0. Then
 1 1 −h h
h, vh

2v2
u
v
−
L
u


1
,
1
1
h(ln v − ln u)
Q(h; u, v) =
(1 − t)u 2 +ht v 2 −ht d t =

0

 1 u,
2
Z
u 6= v,
(2.1)
u = v,
where L(u, v) is the logarithmic mean
v−u
,
L(u, v) = ln v − ln u
u,


u 6= v,
u = v.
Proof. This follows from integration by parts.
3. Main Results
Now we start out to prove some new inequalities of Hermite–Hadamard type for geometrically quasiconvex functions on the co-ordinates.
Theorem 3.1. Let f : ∆ = [a,
b] ×
[c, d] ⊆ R2+ → R be a partial differentiable function on ∆ with a < b,
q
∂2f ∂2f
∈ L1 (∆). If ∂x∂y
c < d, and ∂x∂y
is a geometrically quasi-convex function on the co-ordinates on ∆ for
q ≥ 1, then
1
1
1
1
S(f ) ≤ Q − ; a, b Q − ; c, d + Q − ; a, b Q ; c, d
2
2
2
2
(3.1)
1
1
1
1
+ Q ; a, b Q − ; c, d + Q ; a, b Q ; c, d M (f ),
2
2
2
2
where Q(h; u, v) is defined by (2.1) and
2
∂ f (a, c) ∂ 2 f (a, d) ∂ 2 f (b, c) ∂ 2 f (b, d) .
M (f ) = max ,
,
,
∂x∂y ∂x∂y ∂x∂y ∂x∂y Proof. By Lemma 2.3, we have
2 Z 1Z 1
1−t 1+t 1−λ 1+λ ∂
1−t 1+t
1−λ 1+λ S(f ) ≤
(1 − t)(1 − λ)a 2 b 2 c 2 d 2 f a 2 b 2 , c 2 d 2 d t d λ
∂x∂y
0
0
2 Z 1Z 1
1−t 1+t 1+λ 1−λ ∂
1−t 1+t
1+λ 1−λ +
(1 − t)(1 − λ)a 2 b 2 c 2 d 2 f a 2 b 2 , c 2 d 2 d t d λ
∂x∂y
0
0
2 Z 1Z 1
1+t 1−t 1−λ 1+λ ∂
1+t 1−t
1−λ 1+λ +
(1 − t)(1 − λ)a 2 b 2 c 2 d 2 f a 2 b 2 , c 2 d 2 d t d λ
∂x∂y
0
0
2 Z 1Z 1
1+t 1−t 1+λ 1−λ ∂
1+t 1−t
1+λ 1−λ dtdλ
2
2
2
2
2
2
2
2
+
(1 − t)(1 − λ)a b c
d
d
∂x∂y f a b , c
0
0
(3.2)
(3.3)
, I1 + I2 + I3 + I4 .
2 q
∂ f Using Hölder’s integral inequality, from the co-ordinated geometrically quasi-convexity of ∂x∂y
on ∆, and
by Lemma 2.4, we have
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
1Z 1
Z
I1 ≤
(1 − t)(1 − λ)a
0
1−t
2
0
1Z 1
Z
×
(1 − t)(1 − λ)a
0
0
1Z 1
Z
≤
(1 − t)(1 − λ)a
0
×
Z
(1 − t)(1 − λ)a
0
0
1Z 1
1−t
2
1−t
2
0
1Z 1
Z
b
b
1−t
2
1−t
2
1+t
2
b
1+t
2
1+t
2
b
c
c
1+t
2
1+t
2
(1 − t)(1 − λ)a b c
1
1
= Q − ; a, b Q − ; c, d M (f ).
2
2
=
0
1−λ
2
c
1−λ
2
1−λ
2
c
d
d
1−λ
2
1−λ
2
d
1+λ
2
d
1−1/q
dtdλ
2 q
1/q
∂
1−t 1+t
1−λ 1+λ
2
2
2
2
d
∂x∂y f a b , c
dtdλ
1−1/q
dtdλ
1+λ
2
1+λ
2
d
745
1/q Z
1+λ
2
1+λ
2
dtdλ
0
d t d λ M (f )
1Z 1
M q (f ) d t d λ
1/q
(3.4)
0
0
Using simple techniques of integration shows
1
1
I2 ≤ Q − ; a, b Q ; c, d M (f ),
2
2
1
1
I3 ≤ Q ; a, b Q − ; c, d M (f ),
2
2
and
1
1
I4 ≤ Q ; a, b Q ; c, d M (f ).
2
2
(3.5)
Substituting the inequalities (3.4) to (3.5) into the inequality (3.3) yields (3.1). Theorem 3.1 is proved.
Theorem 3.2. Let f : ∆ = [a,
b] ×
[c, d] ⊆ R2+ → R be a partial differentiable function on ∆ with a < b,
q
∂2f ∂2f
∈ L1 (∆). If ∂x∂y
c < d, and ∂x∂y
is a geometrically quasi-convex function on the co-ordinates on ∆ for
q ≥ 1, then
1/q 1/q
1
1
1
1
2(1/q−1)
q
q
q
q
q
q
q
q
S(f ) ≤ 2
Q − ;a ,b Q − ;c ,d
+ Q − ;a ,b Q ;c ,d
2
2
2
2
(3.6)
1/q 1/q 1 q q
1 q q
1 q q
1 q q
+ Q ;a ,b Q − ;c ,d
+ Q ;a ,b Q ;c ,d
M (f ),
2
2
2
2
where Q(h; u, v) is defined by (2.1) and M (f ) is defined by (3.2).
Proof. Using the inequality
2 q (3.3), by Hölder’s integral inequality, and from the co-ordinated geometrically
∂ f quasi-convexity of ∂x∂y
on ∆, we have
Z
1−1/q
1Z 1
I1 ≤
(1 − t)(1 − λ) d t d λ
0
0
1Z 1
2 1/q
q
∂
1−t 1+t
1−λ 1+λ 2
2
2
2
(1 − t)(1 − λ)a
×
b
c
d
d
∂x∂y f a b , c
dtdλ
0
0
Z 1 Z 1
1/q
1+t
1−λ
1+λ
q 1−t
−2(1−1/q)
q
q
q
(1 − t)(1 − λ)a 2 b 2 c 2 d 2 d t d λ
≤2
M (f )
Z
q 1−t
q 1+t
q 1−λ
q 1+λ
2
2
2
2 0
0
1 q q
1 q q 1/q
−2(1−1/q)
=2
Q − ;a ,b Q − ;c ,d
M (f ).
2
2
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
746
Similarly, we have
1 q q
1 q q 1/q
Q − ;a ,b Q ;c ,d
I2 ≤ 2
M (f ),
2
2
1 q q
1 q q 1/q
−2(1−1/q)
Q ;a ,b Q − ;c ,d
I3 ≤ 2
M (f ),
2
2
−2(1−1/q)
and
I4 ≤ 2
−2(1−1/q)
1 q q
1 q q 1/q
Q ;a ,b Q ;c ,d
M (f ).
2
2
This completes the proof of Theorem 3.2.
2
Theorem 3.3. Let f : ∆ = [a,
2b] ×
q [c, d] ⊆ R+ → R be a partial differentiable function on ∆ with a < b,
2
∂ f ∂ f
c < d, and ∂x∂y
∈ L1 (∆). If ∂x∂y
is a geometrically quasi-convex function on the co-ordinates on ∆ for
q > 1 and q ≥ r ≥ 0, then
S(f ) ≤
1−1/q 1/q
q−r
q−r
q−r
1 q−r
1
1
1
r
r
r
r
Q − ; a q−1 , b q−1 Q − ; c q−1 , d q−1
Q − ;a ,b Q − ;c ,d
2
2
2
2
1−1/q q−r
q−r
1 q−r
1 r r
1 r r 1/q
1 q−r
q−1
q−1
q−1
q−1
Q ;c ,d
Q − ;a ,b Q ;c ,d
+ Q − ;a ,b
2
2
2
2
1−1/q q−r
q−r
1 q−r
1 q−r
1 r r
1 r r 1/q
q−1
q−1
q−1
q−1
+ Q ;a ,b
Q − ;c ,d
Q ;a ,b Q − ;c ,d
2
2
2
2
1−1/q q−r
q−r
1 q−r
1 q−r
1 r r
1 r r 1/q
q−1
q−1
q−1
q−1
+ Q ;a ,b
Q ;c ,d
Q ;a ,b Q ;c ,d
M (f ),
2
2
2
2
where Q(h; u, v) is defined by (2.1) and M (f ) is defined by (3.2).
Proof. Using the inequality
2 q (3.3), by Hölder’s integral inequality, and from the co-ordinated geometrically
∂ f quasi-convexity of ∂x∂y
on ∆, then
Z
1Z 1
I1 ≤
0
1−1/q
1−t 1+t 1−λ 1+λ (q−r)/(q−1)
2
2
2
2
(1 − t)(1 − λ) a b c
d
dtdλ
0
1Z 1
1−t
2
1+t
2
1−λ
2
1+λ
2
0
1Z 1
1−t
2
1+t
2
1−λ
2
1+λ
2
1/q
r ∂ 2 1−t 1+t 1−λ 1+λ q
2
2
2
2
×
(1 − t)(1 − λ) a b c
d
d
dtdλ
∂x∂y f a b , c
0
0
Z 1 Z 1
1−1/q
1−t 1+t 1−λ 1+λ (q−r)/(q−1)
≤
dtdλ
(1 − t)(1 − λ) a 2 b 2 c 2 d 2
Z
0
Z
(1 − t)(1 − λ) a
×
0
b
c
d
r
1/q
dtdλ
M (f )
0
1−1/q q−r
q−r
1 q−r
1 q−r
1 r r
1 r r 1/q
q−1
q−1
q−1
q−1
= Q − ;a ,b
M (f ).
Q − ;c ,d
Q − ;a ,b Q − ;c ,d
2
2
2
2
Similarly, we have
1−1/q q−r
q−r
1 q−r
1 q−r
1 r r
1 r r 1/q
q−1
q−1
q−1
q−1
I2 ≤ Q − ; a , b
Q ;c ,d
Q − ;a ,b Q ;c ,d
M (f ),
2
2
2
2
1−1/q 1/q
q−r
q−r
q−r
1
1 q−r
1
1
r
r
r
r
I3 ≤ Q ; a q−1 , b q−1 Q − ; c q−1 , d q−1
Q ;a ,b Q − ;c ,d
M (f ),
2
2
2
2
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
and
747
1/q
1
1 q−r q−r
1 q−r q−r 1−1/q
1
M (f ).
I4 ≤ Q ; a q−1 , b q−1 Q ; c q−1 , d q−1
Q ; ar , br Q ; cr , dr
2
2
2
2
This completes the required proof.
Remark 3.4. Under the conditions of Theorem 3.3, if r = q, then (3.6) holds.
Corollary 3.5. Under the conditions of Theorem 3.3, when r = 0, we have
1−1/q
q
q
q
q
1
1
−2/q
S(f ) ≤ 2
Q − ; a q−1 , b q−1 Q − ; c q−1 , d q−1
2
2
1−1/q 1−1/q
q
q
q
q
q
q
q
q
1
1
1
1
+ Q − ; a q−1 , b q−1 Q ; c q−1 , d q−1
+ Q ; a q−1 , b q−1 Q − ; c q−1 , d q−1
2
2
2
2
1−1/q q
q
q
q
1
1
M (f ),
+ Q ; a q−1 , b q−1 Q ; c q−1 , d q−1
2
2
where Q(h; u, v) is defined by (2.1) and M (f ) is defined by (3.2).
Theorem 3.6. Let f : ∆ = [a,
b] ×
[c, d] ⊆ R2+ → R be a partial differentiable function on ∆ with a < b,
q
∂2f ∂2f
c < d, and ∂x∂y
∈ L1 (∆). If ∂x∂y
is a geometrically quasi-convex function on the co-ordinates on ∆ for
q > 1, then
2(1−1/q)
q
q
q
q 1/q
1
1
1
1 S(f ) ≤ q − 1
(ac) 2 + (ad) 2 + (bc) 2 + (bd) 2 L a 2 , b 2 L c 2 , d 2
M (f ),
2q − 1
where L(u, v) is the logarithmic mean and M (f ) is defined by (3.2).
Proof. Using the inequality
2 q (3.3), by Hölder’s integral inequality, and from the co-ordinated geometrically
∂ f quasi-convexity of ∂x∂y
on ∆, we have
Z
1Z 1
I1 ≤
q/(q−1)
[(1 − t)(1 − λ)]
0
1−1/q
dtdλ
0
2 1/q
q
∂
1−t 1+t
1−λ 1+λ dtdλ
2 b 2 ,c 2 d 2
×
a
b
c
d
f
a
∂x∂y
0
0
Z 1 Z 1
1/q
q − 1 2(1−1/q)
q 1−t
q 1+t
q 1−λ
q 1+λ
2
2
2
2
a
≤
b
c
d
dtdλ
M (f )
2q − 1
0
0
q
q
q
q 1/q
1
q − 1 2(1−1/q)
(bd) 2 L a 2 , b 2 L c 2 , d 2
M (f ).
=
2q − 1
Z
1Z 1
q 1−t
q 1+t
q 1−λ
q 1+λ
2
2
2
2 Similarly, we have
q−1
2q − 1
2(1−1/q)
q−1
2q − 1
2(1−1/q)
q−1
2q − 1
2(1−1/q)
I2 ≤
I3 ≤
and
I4 ≤
The proof of Theorem 3.6 is complete.
q
q
q
q 1/q
1
(bc) 2 L a 2 , b 2 L c 2 , d 2
M (f ),
q
q
q
q 1/q
1
(ad) 2 L a 2 , b 2 L c 2 , d 2
M (f ),
q
q
q
q 1/q
1
M (f ).
(ac) 2 L a 2 , b 2 L c 2 , d 2
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
748
Theorem 3.7. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R be integrable on ∆ with a < b and c < d. If f is a
geometrically quasi-convex function on the co-ordinates on ∆, then
1
(ln b − ln a)(ln d − ln c)
Z
dZ b
c
a
f (x, y)
d x d y ≤ max{f (a, c), f (a, d), f (b, c), f (b, d)}.
xy
Proof. Letting x = at b1−t and y = cλ d1−λ for t, λ ∈ [0, 1]. By the co-ordinated geometrically quasi-convexity
of f on ∆, we have
1
(ln b − ln a)(ln d − ln c)
dZ b
Z
c
a
f (x, y)
dxdy =
xy
Z
0
Z
1Z 1
f at b1−t , cλ d1−λ d t d λ
0
1Z 1
≤
max{f (a, c), f (a, d), f (b, c), f (b, d)} d t d λ
0
0
= max{f (a, c), f (a, d), f (b, c), f (b, d)}.
This completes the proof of Theorem 3.7.
Theorem 3.8. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R be integrable on ∆ with a < b and c < d. If f is a
geometrically quasi-convex function on the co-ordinates on ∆, then
1
(ln b − ln a)(ln d − ln c)
dZ b
Z
f (x, y) d x d y ≤ L(a, b)L(c, d) max{f (a, c), f (a, d), f (b, c), f (b, d)},
c
a
where L(u, v) is the logarithmic mean.
Proof. Similarly as in Theorem 3.7, by the co-ordinated geometrically quasi-convexity of f on ∆, we have
1
(ln b − ln a)(ln d − ln c)
Z
dZ b
Z
1Z 1
at b1−t cλ d1−λ f at b1−t , cλ d1−λ d t d λ
0
0
Z 1Z 1
≤ max{f (a, c), f (a, d), f (b, c), f (b, d)}
at b1−t cλ d1−λ d t d λ
f (x, y) d x d y =
c
a
0
0
= L(a, b)L(c, d) max{f (a, c), f (a, d), f (b, c), f (b, d)}.
The proof of Theorem 3.8 is complete.
We proceed similarly as in the proof of Theorems 3.7 and 3.8, we can obtain the following theorem.
Theorem 3.9. Let f, g : ∆ = [a, b] × [c, d] ⊆ R2+ → R0 = [0, ∞) be integrable on ∆ with a < b and c < d. If
f, g are geometrically quasi-convex functions on the co-ordinates on ∆, then
1
(ln b − ln a)(ln d − ln c)
Z
c
dZ b
a
f (x, y)g(x, y)
dxdy
xy
≤ max{f (a, c), f (a, d), f (b, c), f (b, d)} max{g(a, c), g(a, d), g(b, c), g(b, d)}
and
Z dZ b
1
f (x, y)g(x, y) d x d y
(ln b − ln a)(ln d − ln c) c a
≤ [L(a, b)L(c, d)]2 max{f (a, c), f (a, d), f (b, c), f (b, d)} max{g(a, c), g(a, d), g(b, c), g(b, d)},
where L(u, v) is the logarithmic mean.
X.-Y. Guo, F. Qi, B.-Y. Xi, J. Nonlinear Sci. Appl. 8 (2015), 740–749
749
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China under Grant
No. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner
Mongolia Autonomous Region under Grant Grant No. NJZY14192, by the Science Research Fund of the
Inner Mongolia University for Nationalities under Grant No. NMD1302, and by the Scientific Innovation
Project for Graduates at the Inner Mongolia University for Nationalities under Grant No. NMDSS1419,
China.
The authors thank anonymous referees for their valuable comments on and careful corrections to the
original version of this paper.
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