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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 214123, 8 pages
http://dx.doi.org/10.1155/2013/214123
Research Article

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.

Abstract

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.

1. Introduction

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, [114] 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 [8].

The following theorem has been given in [4].

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 [9]. 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 [15].

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

The following cases from the above inequalities are of interest [6, 7].

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 [6].

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

Remark 5. Note that if we substitute , , , , and in Theorem 2, we get the following result of Dragomir [9] for convex function defined on the interval and ,

The following refinement of Hölder inequality holds.

Corollary 6. Let and be two positive n-tuples. Then for , , , one has

Proof. Using the functions,, , andin (28), we get (29).

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.

Acknowledgments

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).

References

  1. M. Adil Khan, M. Anwar, J. Jakšetić, and J. Pečarić, “On some improvements of the Jensen inequality with some applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 323615, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Adil Khan, S. Khalid, and J. Pečarić, “Improvement of Jensen's inequality in terms of Gâteaux derivatives for convex functions in linear spaces with applications,” Kyungpook Mathematical Journal, vol. 52, no. 4, pp. 495–511, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  3. M. A. Khan, S. Khalid, and J. Pečarić, “Refinements of some majorization type inequalities,” Journal of Mathematical Inequalities, vol. 7, no. 1, pp. 73–92, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Klaričić Bakula and J. Pečarić, “On the Jensen's inequality for convex functions on the co-ordinates in a rectangle from the plane,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1271–1292, 2006. View at MathSciNet
  5. L. Horváth, “A parameter-dependent refinement of the discrete Jensen's inequality for convex and mid-convex functions,” Journal of Inequalities and Applications, vol. 2011, article 26, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. S. Dragomir, “A refinement of Jensen’s inequality with applications for f-divergence measures,” Taiwanese Journal of Mathematics, vol. 14, no. 1, pp. 153–164, 2010.
  7. S. S. Dragomir, “Some refinements of Jensen's inequality,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 518–522, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. S. Dragomir, “On Hadamard's inequality for the convex mappings defined on a ball in the space and applications,” Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 177–187, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. S. Dragomir, “A new refinement of Jensen's inequality in linear spaces with applications,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1497–1505, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Rooin, “Some refinements of discrete Jensen's inequality and some of its applications,” Nonlinear Functional Analysis and Applications, vol. 12, no. 1, pp. 107–118, 2007. View at Zentralblatt MATH · View at MathSciNet
  11. L.-C. Wang, X.-F. Ma, and L.-H. Liu, “A note on some new refinements of Jensen's inequality for convex functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 2, article 48, 2009. View at Zentralblatt MATH · View at MathSciNet
  12. X. L. Tang and J. J. Wen, “Some developments of refinned Jensen's inequality,” Journal of Southwest University for Nationalities, vol. 29, pp. 20–26, 2003.
  13. S. Hussain and J. Pečarić, “An improvement of Jensen's inequality with some applications,” Asian-European Journal of Mathematics, vol. 2, no. 1, pp. 85–94, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. Zabandan and A. Kılıçman, “A new version of Jensen's inequality and related results,” Journal of Inequalities and Applications, vol. 2012, article 238, 2012. View at Publisher · View at Google Scholar
  15. G. Zabandan and A. Kılıçman, “Several integral inequalities and an upper bound for the bidimensional Hermite-Hadamard inequality,” Journal of Inequalities and Applications, vol. 2013, article 27, 2013. View at Publisher · View at Google Scholar