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On The Hadamard's Inequality for Log-Convex Functions on the Coordinates
Journal of Inequalities and Applications volume 2009, Article number: 283147 (2009)
Abstract
Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given.
1. Introduction
Let be a convex mapping defined on the interval
of real numbers and
, with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ1_HTML.gif)
holds, this inequality is known as the Hermite-Hadamard inequality. For refinements, counterparts, generalizations and new Hadamard-type inequalities, see [1–8].
A positive function is called log-convex on a real interval
, if for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ2_HTML.gif)
If is a positive log-concave function, then the inequality is reversed. Equivalently, a function
is log-convex on
if
is positive and
is convex on
. Also, if
and
exists on
, then
is log-convex if and only if
.
The logarithmic mean of the positive real numbers ,
, is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ3_HTML.gif)
A version of Hadamard's inequality for log-convex (concave) functions was given in [9], as follows.
Theorem 1.1.
Suppose that is a positive log-convex function on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ4_HTML.gif)
If is a positive log-concave function, then the inequality is reversed.
For refinements, counterparts and generalizations of log-convexity see [9–13].
A convex function on the coordinates was introduced by Dragomir in [8]. A function which is convex in
is called coordinated convex on
if the partial mapping
,
and
,
, are convex for all
and
.
An inequality of Hadamard's type for coordinated convex mapping on a rectangle from the plane established by Dragomir in [8], is as follows.
Theorem 1.2.
Suppose that is coordinated convex on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ5_HTML.gif)
The above inequalities are sharp.
The maximum modulus principle in complex analysis states that if is a holomorphic function, then the modulus
cannot exhibit a true local maximum that is properly within the domain of
. Characterizations of the maximum principle for sub(super)harmonic functions are considered in [14], as follows.
Theorem 1.3.
Let be a region and let
be a sub(super)harmonic function. If there is a point
with
, for all
then
is a constant function.
Theorem 1.4.
Let be a region and let
and
be bounded real-valued functions defined on
such that
is subharmonic and
is superharmonic. If for each point
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ6_HTML.gif)
then for all
or
and
is harmonic.
In this paper, a new version of the maximum (minimum) principle in terms of convexity, and some inequalities of the Hadamard type are obtained.
2. On Coordinated Convexity and Sub(Super)Harmonic Functions
Consider the -dimensional interval
in
. A function
is called convex in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ7_HTML.gif)
holds for all and
.
As in [8], we define a log-convex function on the coordinates as follows: a function will be called coordinated log-convex on
if the partial mappings
,
and
,
, are log-convex for all
and
. A formal definition of a coordinated log-convex function may be stated as follows.
Definition 2.1.
A function will be called coordinated log-convex on
, for all
and
, if the following inequality holds,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ8_HTML.gif)
Equivalently, we can determine whether or not the function is coordinated log-convex by using the following lemma.
Lemma 2.2.
Let . If
is twice differentiable then
is coordinated log-convex on
if and only if for the functions
, defined by
and
, defined by
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ9_HTML.gif)
Proof.
The proof is straight forward using the elementary properties of log-convexity in one variable.
Proposition 2.3.
Suppose that is twice differentiable on
and log-convex on
and
is twice differentiable on
and log-convex on
. Let
be a twice differentiable function defined by
, then
is coordinated log-convex on
.
Proof.
This follows directly using Lemma 2.2.
The following result holds.
Proposition 2.4.
Every log-convex function is log-convex on the coordinates, but the converse is not generally true.
Proof.
Suppose that is convex in
. Consider the function
,
, then for
, and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ10_HTML.gif)
which shows the log-convexity of . The proof that
,
, is also log-convex on
for all
follows likewise. Now, consider the mapping
given by
. It is obvious that
is log-convex on the coordinates but not log-convex on
. Indeed, if
and
, we have:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ11_HTML.gif)
Thus, for all and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ12_HTML.gif)
which shows that is not log-convex on
.
In the following, a Jensen-type inequality for coordinated log-convex functions is considered.
Proposition 2.5.
Let be a positive coordinated log-convex function on the open set
and let
,
. If
and
,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ13_HTML.gif)
Proof.
Let ,
be such that
, and let
,
be such that
, then we have,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ14_HTML.gif)
and, since is positive,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ15_HTML.gif)
which is as required.
Remark 2.6.
Let , then the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ16_HTML.gif)
The above result may be generalized to the integral form as follows.
Proposition 2.7.
Let be a positive coordinated log-convex function on the
and let
be integrable with
, and let
be integrable with
. If
is positive,
, and
is integrable on
and
is positive,
, and
is integrable on
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ17_HTML.gif)
Proof.
Applying Jensen's integral inequality in one variable on the -coordinate and on the
-coordinate we get the required result. The details are omitted.
Theorem 2.8.
Let be a positive coordinated log-convex function in
, then for all distinct
, such that
and distinct
such that
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ18_HTML.gif)
Proof.
Let be distinct points in
and let
be distinct points in
. Setting
,
and let
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ19_HTML.gif)
and we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ20_HTML.gif)
From this inequality it is easy to deduce the required result (2.12).
The subharmonic functions exhibit many properties of convex functions. Next, we give some results for the coordinated convexity and sub(super)harmonic functions.
Proposition 2.9.
Let be coordinated convex (concave) on
. If
is a twice differentiable on
, then
is sub(super)harmonic on
.
Proof.
Since is coordinated convex on
then the partial mappings
,
and
,
, are convex for all
and
. Equivalently, since
is differentiable we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ21_HTML.gif)
for all , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ22_HTML.gif)
for all , which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ23_HTML.gif)
which shows that is subharmonic. If
is coordinated concave on
, replace "
'' by "
'' above, we get that
is superharmonic on
.
We now give two version(s) of the Maximum (Minimum) Principle theorem using convexity on the coordinates.
Theorem 2.10.
Let be a coordinated convex (concave) function on
. If
is twice differentiable in
and there is a point
with
, for all
then
is a constant function.
Proof.
By Proposition 2.9, we get that is sub(super)harmonic. Therefore, by Theorem 1.3 and the maximum principal the required result holds (see [14, page 264]).
Theorem 2.11.
Let and
be two twice differentiable functions in
. Assume that
and
are bounded real-valued functions defined on
such that
is coordinated convex and
is coordinated concave. If for each point
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ24_HTML.gif)
then for all
or
and
is harmonic.
Proof.
By Proposition 2.9, we get that is subharmonic and
is superharmonic. Therefore, by Theorem 1.4 and using the maximum principal the required result holds, (see [14, page 264]).
Remark 2.12.
The above two results hold for log-convex functions on the coordinates, simply, replacing by
, to obtain the results.
3. Some Inequalities and Applications
In the following we develop a Hadamard-type inequality for coordinated log-convex functions.
Corollary 3.1.
Suppose that is log-convex on the coordinates of
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ25_HTML.gif)
For a positive coordinated log-concave function , the inequalities are reversed.
Proof.
In Theorem 1.2, replace by
and we get the required result.
Lemma 3.2.
For with
, the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ26_HTML.gif)
is convex for all . Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ27_HTML.gif)
for all .
Proof.
Since is twice differentiable for all
with
, we note that for all
,
, which shows that
is increasing and thus
is nonnegative which is equivalent to saying that
is increasing and hence
is convex. Now, using inequality (1.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ28_HTML.gif)
which completes the proof.
Theorem 3.3.
Suppose that is log-convex on the coordinates of
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ29_HTML.gif)
then the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ30_HTML.gif)
hold, where is the Euler constant,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ31_HTML.gif)
is the exponential integral function. For a coordinated log-concave function , the inequalities are reversed.
Proof.
Since is log-convex on the coordinates of
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ32_HTML.gif)
Integrating the previous inequality with respect to and
on
, we have,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ33_HTML.gif)
Therefore, by (3.9) and for nonzero, positive , we have the following cases.
(1)If , the result is trivial.
(2)If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ34_HTML.gif)
-
(3)
If
, then
(3.11)
-
(4)
If
, then
(3.12)
-
(5)
If
, then
(3.13)
-
(6)
If
, then
(3.14)
-
(7)
If
, then
(3.15)
-
(8)
If
then
(3.16)
Therefore, by Lemma 3.2, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ41_HTML.gif)
-
(9)
If
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ42_HTML.gif)
which is difficult to evaluate because it depends on the values of and
.
Remark 3.4.
The integrals in (3), (4), and (7) in the proof of Theorem 2.11 are evaluated using Maple Software.
Corollary 3.5.
In Theorem 3.3, if
(1), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ43_HTML.gif)
and for instance, if ,
we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ44_HTML.gif)
(2), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ45_HTML.gif)
and for instance, if , ?
, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F283147/MediaObjects/13660_2009_Article_1929_Equ46_HTML.gif)
Proof.
Follows directly by applying inequality (1.4).
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Acknowledgment
The authors acknowledge the financial support of the Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM–GUP–TMK–07–02–107).
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Alomari, M., Darus, M. On The Hadamard's Inequality for Log-Convex Functions on the Coordinates. J Inequal Appl 2009, 283147 (2009). https://doi.org/10.1155/2009/283147
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DOI: https://doi.org/10.1155/2009/283147