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On The Hadamard's Inequality for LogConvex 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 logconvex 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
holds, this inequality is known as the HermiteHadamard inequality. For refinements, counterparts, generalizations and new Hadamardtype inequalities, see [1–8].
A positive function is called logconvex on a real interval , if for all and ,
If is a positive logconcave function, then the inequality is reversed. Equivalently, a function is logconvex on if is positive and is convex on . Also, if and exists on , then is logconvex if and only if .
The logarithmic mean of the positive real numbers , , is defined as
A version of Hadamard's inequality for logconvex (concave) functions was given in [9], as follows.
Theorem 1.1.
Suppose that is a positive logconvex function on , then
If is a positive logconcave function, then the inequality is reversed.
For refinements, counterparts and generalizations of logconvexity 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
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 realvalued functions defined on such that is subharmonic and is superharmonic. If for each point
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
holds for all and .
As in [8], we define a logconvex function on the coordinates as follows: a function will be called coordinated logconvex on if the partial mappings , and , , are logconvex for all and . A formal definition of a coordinated logconvex function may be stated as follows.
Definition 2.1.
A function will be called coordinated logconvex on , for all and , if the following inequality holds,
Equivalently, we can determine whether or not the function is coordinated logconvex by using the following lemma.
Lemma 2.2.
Let . If is twice differentiable then is coordinated logconvex on if and only if for the functions , defined by and , defined by , we have
Proof.
The proof is straight forward using the elementary properties of logconvexity in one variable.
Proposition 2.3.
Suppose that is twice differentiable on and logconvex on and is twice differentiable on and logconvex on . Let be a twice differentiable function defined by , then is coordinated logconvex on .
Proof.
This follows directly using Lemma 2.2.
The following result holds.
Proposition 2.4.
Every logconvex function is logconvex on the coordinates, but the converse is not generally true.
Proof.
Suppose that is convex in . Consider the function , , then for , and , we have
which shows the logconvexity of . The proof that , , is also logconvex on for all follows likewise. Now, consider the mapping given by . It is obvious that is logconvex on the coordinates but not logconvex on . Indeed, if and , we have:
Thus, for all and , we have
which shows that is not logconvex on .
In the following, a Jensentype inequality for coordinated logconvex functions is considered.
Proposition 2.5.
Let be a positive coordinated logconvex function on the open set and let , . If and , , then
Proof.
Let , be such that , and let , be such that , then we have,
and, since is positive,
which is as required.
Remark 2.6.
Let , then the following inequality holds:
The above result may be generalized to the integral form as follows.
Proposition 2.7.
Let be a positive coordinated logconvex 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
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 logconvex function in , then for all distinct , such that and distinct such that , the following inequality holds:
Proof.
Let be distinct points in and let be distinct points in . Setting , and let , , we have
and we can write
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
for all , and
for all , which imply that
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 realvalued functions defined on such that is coordinated convex and is coordinated concave. If for each point
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 logconvex functions on the coordinates, simply, replacing by , to obtain the results.
3. Some Inequalities and Applications
In the following we develop a Hadamardtype inequality for coordinated logconvex functions.
Corollary 3.1.
Suppose that is logconvex on the coordinates of , then
For a positive coordinated logconcave 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
is convex for all . Moreover,
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
which completes the proof.
Theorem 3.3.
Suppose that is logconvex on the coordinates of . Let
then the inequalities
hold, where is the Euler constant,
is the exponential integral function. For a coordinated logconcave function , the inequalities are reversed.
Proof.
Since is logconvex on the coordinates of , then
Integrating the previous inequality with respect to and on , we have,
Therefore, by (3.9) and for nonzero, positive , we have the following cases.
(1)If , the result is trivial.
(2)If , then

(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

(9)
If , we have
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
and for instance, if , we deduce
(2), then
and for instance, if , ? , we deduce
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 LogConvex Functions on the Coordinates. J Inequal Appl 2009, 283147 (2009). https://doi.org/10.1155/2009/283147
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Keywords
 Harmonic Function
 Holomorphic Function
 Distinct Point
 Require Result
 Subharmonic Function