- Research Article
- Open Access
On The Hadamard's Inequality for Log-Convex Functions on the Coordinates
© M. Alomari and M. Darus. 2009
- Received: 15 January 2009
- Accepted: 20 July 2009
- Published: 3 August 2009
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.
- Harmonic Function
- Holomorphic Function
- Distinct Point
- Require Result
- Subharmonic Function
Let be a convex mapping defined on the interval of real numbers and , with , then
A positive function is called log-convex on a real interval , if for all and ,
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
A version of Hadamard's inequality for log-convex (concave) functions was given in , as follows.
If is a positive log-concave function, then the inequality is reversed.
A convex function on the coordinates was introduced by Dragomir in . 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 , is as follows.
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 , as follows.
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.
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.
Consider the -dimensional interval in . A function is called convex in if
holds for all and .
As in , 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.
Equivalently, we can determine whether or not the function is coordinated log-convex by using the following lemma.
The proof is straight forward using the elementary properties of log-convexity in one variable.
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 .
This follows directly using Lemma 2.2.
The following result holds.
Every log-convex function is log-convex on the coordinates, but the converse is not generally true.
which shows that is not log-convex on .
In the following, a Jensen-type inequality for coordinated log-convex functions is considered.
which is as required.
The above result may be generalized to the integral form as follows.
Applying Jensen's integral inequality in one variable on the -coordinate and on the -coordinate we get the required result. The details are omitted.
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.
Let be coordinated convex (concave) on . If is a twice differentiable on , then is sub(super)harmonic on .
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.
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.
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]).
then for all or and is harmonic.
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]).
The above two results hold for log-convex functions on the coordinates, simply, replacing by , to obtain the results.
In the following we develop a Hadamard-type inequality for coordinated log-convex functions.
For a positive coordinated log-concave function , the inequalities are reversed.
In Theorem 1.2, replace by and we get the required result.
for all .
which completes the proof.
is the exponential integral function. For a coordinated log-concave function , the inequalities are reversed.
Therefore, by (3.9) and for nonzero, positive , we have the following cases.
(1)If , the result is trivial.
(2)If , then
If , we have
which is difficult to evaluate because it depends on the values of and .
The integrals in (3), (4), and (7) in the proof of Theorem 2.11 are evaluated using Maple Software.
In Theorem 3.3, if
and for instance, if , we deduce
and for instance, if , ? , we deduce
Follows directly by applying inequality (1.4).
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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