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.
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 .
A version of Hadamard's inequality for log-convex (concave) functions was given in , as follows.
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.
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
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.
The proof is straight forward using the elementary properties of log-convexity in one variable.
This follows directly using Lemma 2.2.
The following result holds.
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.
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.
We now give two version(s) of the Maximum (Minimum) Principle theorem using convexity on the coordinates.
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]).
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]).
3. Some Inequalities and Applications
In the following we develop a Hadamard-type inequality for coordinated log-convex functions.
which completes the proof.
The integrals in (3), (4), and (7) in the proof of Theorem 2.11 are evaluated using Maple Software.
In Theorem 3.3, if
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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