ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES†

Journal of Applied Mathematics & Informatics.
2014.
Sep,
32(3_4):
303-314

- Received : June 17, 2013
- Accepted : September 16, 2013
- Published : September 28, 2014

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In this paper, a new lemma is established and several new inequalities for differentiable co-ordinated quasi-convex functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated quasi-convex functions in two variables are obtained.
AMS Mathematics Subject Classification : 26A51, 26D15.
f
:
I
⊆ ℝ → ℝ be a convex function and
a, b
∈
I
with
a
<
b
, we have the following double inequality
This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping.
Definition 1.1.
A function
f
: [
a, b
] →
R
is said to be quasi-convex on [
a, b
], if
holds for all
x, y
∈ [
a, b
] and
λ
∈ [0, 1].
Clearly, any convex function is a quasi-convex function, but the converse is not generally true.
In
[4]
, S. S. Dragomir defined convex functions on the co-ordinates as following:
Let us consider the bidimensional interval Δ := [
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
holds for all (
x, y
), (
z, w
) ∈ Δ and
λ
∈ [0, 1].
A function
f
: Δ → ℝ is said to be co-ordinated 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 co-ordinated convex functions may be stated as follows:
Definition 1.2.
A function
f
: Δ → ℝ is said to be convex on co-ordinates on Δ if the inequality
holds for all (
x, y
), (
z, y
), (
x, w
), (
z, w
) ∈ Δ and
t
,
λ
∈ [0, 1].
S. S. Dragomir in
[4]
established the following Hadamard-type inequalities for co-ordinated convex functions in a rectangle from the plane ℝ
^{2}
.
Theorem 1.3.
Suppose that f
: Δ = [
a, b
] × [
c, d
] → ℝ
is convex on the coordinates on
Δ.
Then one has the inequalities:
The concept of quasi-convex function on the co-ordinates was introduced by
Ö
zdemir et al. in (
[9]
, 2012).
Let us consider the bidimensional interval Δ := [
a, b
]×[
c, d
] in ℝ
^{2}
with
a
<
b
and
c
<
d
, a mapping
f
: Δ → ℝ is said to be a quasi-convex function on Δ if the inequality
holds for all (
x, y
), (
z, w
) ∈ Δ and
λ
[0, 1].
A function
f
: Δ → ℝ is said to be quasi-convex functions on the co-ordinates 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 of quasi-convex functions on the co-ordinates as follows:
Definition 1.4.
A function
f
: Δ → ℝ is said to be a quasi-convex function on the co-ordinates on Δ if the inequality
holds for all (
x, y
), (
z, y
), (
x, w
), (
z, w
) ∈ Δ with
t
,
λ
∈ [0, 1].
In (
[10]
, 2012), M. Z. Sarıkaya et al. established some inequalities for coordinated convex functions based on the following lemma.
Lemma 1.5.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
then the following equality holds:
In (
[7]
, 2012), M. E.
Ö
zdemir et al. established the following inequalities for quasi-convex functions on the co-ordinates based on Lemma 1.5.
Theorem 1.6.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ = [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is a quasi-convex function on the co-ordinates on
Δ,
then the following inequality holds:
where
Theorem 1.7.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ = [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is a quasi-convex function on the co-ordinates on
Δ
and q
> 1,
then:
where A is defined in Theorem 1.6 and
Some new integral inequalities that are related to the Hermite-Hadamard type for co-ordinated convex functions are also established by many authors.
In (
[1]
,
[2]
, 2008), M. Alomari and M. Darus defined co-ordinated
s
-convex functions and proved some inequalities based on this definition. In (
[5]
, 2009), M. A. Latif and M. Alomari defined co-ordinated
h
-convex functions and proved some inequalities based on this definition. In (
[3]
, 2009), Alomari et al. established some Hadamard-type inequalities for coordinated log-convex functions.
In (
[6]
, 2012), M. A. Latif and S. S. Dragomir obtained some new Hadamard type inequalities for differentiable co-ordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated convex functions in two variables based on the following lemma:
Theorem 1.8.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
then the following equality holds:
where
Theorem 1.9
(
[6]
).
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is convex on the co-ordinates on
Δ,
then the following inequality holds:
where
Theorem 1.10
(
[6]
).
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is convex on the co-ordinates on
Δ
and
then the following inequality holds:
where A is as given in Theorem 1.9.
For recent results and generalizations concerning Hermite-Hadamard type inequality for differentiable co-ordinated convex functions see (
[8]
, 2012) and the references given therein.
In this paper, we establish several new inequalities for differentiable co-ordinated quasi-convex functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated quasi-convex functions in two variables.
Lemma 2.1.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
then the following equality holds:
where
Proof
. Since
Thus, by integration by parts, it follows that
Similarly, we can get
and
Now
Multiplying the both sides by
and using Lemma 1.8, which completes the proof.
Theorem 2.2.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is a quasi-convex function on the co-ordinates on
Δ,
then the following inequality holds
:
where
Proof
. From Lemma 2.1, we obtain
Because
is quasi-convex on the co-ordinates on Δ, then one has
On the other hand, we have
The proof is completed.
The corresponding version for powers of the absolute value of the fourth partial derivative is incorporated in the following theorems.
Theorem 2.3.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is a quasi-convex function on the co-ordinates on
Δ
and q
> 1,
then:
where A is defined in Theorem 3.1 and
Proof
. From Lemma 2.1, we obtain
By using the well known H
ö
lder’s inequality for double integrals, then one has
Because
is quasi-convex on the co-ordinates on Δ, then one has
We note that
Hence, it follows that
So, the proof is completed.
Theorem 2.4.
Let f
: Δ ⊆ ℝ
^{2}
→ ℝ
be a partial differentiable mapping on
Δ := [
a, b
] × [
c, d
]
in
ℝ
^{2}
with a
<
b and c
<
d
.
is a quasi-convex function on the co-ordinates on
Δ
and q
> 1,
then:
where A is defined in Theorem 3.1.
Proof
. From Lemma 2.1, we obtain
By using the well known power mean inequality for double integrals, then one has
Because
is quasi-convex on the co-ordinates on Δ, then one has
Thus, it follows that
Thus, we get the following inequality
which complete the proof.
Feixiang Chen received his MS degree from Xi’an Jiaotong University. Since 2009 he has been teaching at Chongqing Three Gorges University. His research interests include integral inequalities on several kinds of convex functions and applications.
School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, 404000, P.R. China.
e-mail: cfx2002@126.com

Hermite-Hadamard’s inequality
;
quasi-convex functions on the co-ordinates
;
Hölder’s inequality
;
power mean inequality

1. Introduction

Let
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2. Main results

To establishing our results, we need the following lemma.
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BIO

Alomari M.
,
Darus M.
2008
The Hadamard’s inequality for s-convex functions of 2-variables on the co-ordinates.
Int. Journal of Math. Analysis
2
(13)
629 -
638

Alomari M.
,
Darus M.
2008
Hadamard-Type inequalities for s-convex functions.
Int. Math. Forum
3
(40)
1965 -
1975

Alomari M.
,
Darus M.
2009
On the Hadamard’s inequality for log-convex functions on the coordinates.
J. Inequal. Appl
2009
283147 -
** DOI : 10.1155/2009/283147**

Dragomir S. S.
2001
On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane.
Taiwanese J. Math
5
(4)
775 -
788

Latif M. A.
,
Alomari M.
2009
On Hadamard-type inequalities for h-convex functions on the co-ordinates.
Int. Journal of Math. Analysis
3
(33)
1645 -
1656

Latif M. A.
,
Dragomir S. S.
2012
On some new inequalities for differentiable co-ordinated convex functions.
J. Inequal. Appl
2012
28 -
** DOI : 10.1186/1029-242X-2012-28**

Özdemir M. E.
,
Yıldız Çetin
,
Akdemir A. O.
2012
On some new Hadamard-type inequalities for co-ordinated quasi-convex functions.
Hacet. J. Math. Stat
41
(5)
697 -
707

Özdemir M. E.
,
Kavurmaci H.
,
Akdemir A. O.
,
Avci M.
2012
Inequalities for convex and s-convex functions on Δ = [a, b] × [c, d].
J. Inequal. Appl
2012
20 -

Özdemir M. E.
,
Akdemir A. O.
,
Yıldız Çetin
2012
On co-ordinated quasi-convex functions.
Czechoslovak Math. J.
62
(4)
889 -
900
** DOI : 10.1007/s10587-012-0072-z**

Sarıkaya M. Z.
,
Özdemir M. E.
,
Dragomir S. S.
2012
New some Hadamard’s type inequalities for co-ordinated convex functions.
Tamsui Oxf. J. Inf. Math. Sci
28
(2)
137 -
152

Citing 'ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES†
'

@article{ E1MCA9_2014_v32n3_4_303}
,title={ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES†}
,volume={3_4}
, url={http://dx.doi.org/10.14317/jami.2014.303}, DOI={10.14317/jami.2014.303}
, number= {3_4}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={Chen, Feixiang}
, year={2014}
, month={Sep}