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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 214123, 8 pages
Further Refinements of Jensen’s Type Inequalities for the Function Defined on the Rectangle
1Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2Institute of Mathematics, National University of Mongolia, P.O. Box 46A/104, 14201 Ulaanbaatar, Mongolia
3Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Received 26 September 2013; Accepted 10 November 2013
Academic Editor: Abdullah Alotaibi
Copyright © 2013 M. Adil Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We give refinement of Jensen’s type inequalities given by Bakula and Pečarić (2006) for the co-ordinate convex function. Also we establish improvement of Jensen’s inequality for the convex function of two variables.
Jensen’s inequality for convex functions plays a crucial role in the theory of inequalities due to the fact that other inequalities such as the arithmetic mean-geometric mean inequality, the Hölder and Minkowski inequalities, and the Ky Fan inequality, can be obtained as particular cases of it. Therefore, it is worth studying it thoroughly and refining it from different point of view. There are many refinements of Jensen’s inequality; see, for example, [1–14] and the references in them.
A function, withand is called convex on the co-ordinates if the partial mappings defined as anddefined as are convex for all,. Note that every convex function is co-ordinate convex, but the converse is not generally true .
The following theorem has been given in .
Theorem 1. Let be a convex function on the co-ordinates on. If is an n-tuple in, is m-tuple in, is a nonnegative n-tuple, and is a nonnegative m-tuple such thatand, then where, and.
Recently Dragomir has given new refinement for Jensen inequality in . The purpose of this paper is to give related refinements of Jensen’s type inequalities (1) for the co-ordinate convex function. We will also discuss some particular interesting cases. We establish improvement of Jensen’s inequality for the convex function defined on the rectangles. For related improvements of Jensen’s inequality, see, for example, [1, 2, 9, 13, 14]. For further several related integral inequalities, see .
2. Main Results
Let be convex on the co-ordinate on. If,, ,,with, and, then for any subsetsand, we assume thatand. Define, , , and. For the functionand the-, -tuples,,, , and , we define the following functionals: where, and .
It is worth to observe that for, , and, , we have the functionals The following refinement of (1) holds.
Theorem 2. Let be a co-ordinate convex function on. If,, , ,, with , and , then for any subsets and , one has where, and .
Proof. One-dimensional Jensen’s inequality gives us
As we have
so by Jensen’s inequality, we have
As the functionis convex on the first co-ordinate, so we have
Now, from (7) and (8), we have
Similarly, we can write
Multiplying (9) and (10), respectively, byandand summing overand, we obtain
Adding (11) and (12), we have
Again by one-dimensional Jensen’s inequality, we have
As we have the functional
so by Jensen’s inequality, we get
and as the functionis convex on the first co-ordinate, so we have
Now from (16) and (17), we have
Similarly, we can prove that
Adding (18) and (19), we get
Combining (13) and (20), we have
Remark 3. We observe that the inequalities in (4) can be written equivalently as These inequalities imply the following results:
Moreover, from the above, we also have
We discuss the following particular cases of the above inequalities which is of interest .
In the case when andforand, consider the natural numbers,with andand define We can give the following result.
Corollary 4. Let be a co-ordinate convex function on. If and , then for anyand, one has
In particular, we have the bounds
The following refinement of Hölder inequality holds.
Corollary 6. Let and be two positive n-tuples. Then for , , , one has
Remark 7. As mentioned above from the inequalities in (29), we can write
The following improvement of Jensen’s inequality is valid.
Theorem 8. Let be convex on the co-ordinates of. Ifis an n-tuple in, is an m-tuple in, is a nonnegative n-tuple such that, and is a nonnegative m-tuple such that, then where, and.
Proof. Sinceis convex on, therefore we have From the above inequality, we have Let,,, and, then (33) becomes Multiplying (34) by and and summing overand, we have One has Therefore (35) becomes Multiplying both hand sides by , we have This completes the proof.
Conflict of Interests
The authors declare that they have no conflict of interests regarding publication of this paper.
The authors are grateful to the referees for the useful comments regarding presentation in the early version of the paper. The last author also acknowledges that the present work was partially supported by the University Putra Malaysia (UPM).
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