### More About Petrovic’s Inequality on Coordinates via m-Convex Functions and Related Results

Download PDF

**Authors:**A. UR REHMAN, G. FARID AND W. IQBAL

**DOI:**10.46793/KgJMat2003.335R

**Abstract:**

In this paper the authors extend Petrović’s inequality for coordinated m-convex functions in the plane and also ﬁnd Lagrange type and Cauchy type mean value theorems for Petrović’s inequality for m-convex functions and coordinated m-convex functions. The authors consider functional due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-m-convex functions.

**Keywords:**

Petrović’s inequality, mean value theorem, log-convexity, m-convex functions on coordinates.

**References:**

[1] M. Alomari and M. Darus, On the Hadamard’s inequality for log convex functions on coordinates, J. Inequal. Appl. 2009(1) (2009), 13 pages.

[2] M. K. Bakula, J. Pečarić and M. Ribičić, Companion inequalities to Jensen’s inequality for m-convex and (α,m)-convex functions, Journal of Inequalities in Pure and Applied Mathematics 7(5) (2006), 32 pages.

[3] P. S. Bullen, Handbook of Means and Their Inequalities, Springer Science & Business Media, Dordrecht, Boston, London, 2013.

[4] S. Butt, J. Pečarić and A. U. Rehman, Exponential convexity of Petrović and related functional, J. Inequal. Appl. 2011(1) (2011), 16 pp.

[5] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m–convex functions, Tamkang J. Math. 33(1) (2002), 45–56.

[6] S. S Dragomir, On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math. 5(4) (2001), 775–788.

[7] S. S. Dragomir, Charles E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, Melbourne, 2000.

[8] G. Farid, M. Marwan and A. U. Rehman, New mean value theorems and generalization of Hadamard inequality via coordinated m-convex functions, J. Inequal. Appl. 2015(1) (2015), 11 pages.

[9] M. Krnić, R. Mikić and J. Pečarić, Strengthened converses of the Jensen and Edmundson-Lah-Ribarič inequalities, Adv. Oper. Theory 1(1) (2016), 104–122.

[10] T. Lara, E. Rosales and J. L. Sánchez, New properties of m-convex functions, Int. J. Appl. Math. Anal. Appl. 9(15) (2015), 735–742.

[11] V. G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, Romania, 1993.

[12] Z. Pavić and M. A. Ardiç, The most important inequalities of m-convex functions, Turkish J. Math. 41(3) (2017), 625–635.

[13] J. E. Pečarić, On the Petrović’s inequality for convex functions, Glas. Mat. 18(38) (1983), 77–85.

[14] J. Pečarić and V. Čuljak, Inequality of Petrović and Giaccardi for convex function of higher order, Southeast Asian Bull. Math. 26(1) (2003), 57–61.

[15] J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1991.

[16] J. Pečarić and A. U. Rehman, On logarithmic convexity for power sums and related results, J. Inequal. Appl. 2008(1) (2008), 10 pages.

[17] M. Petrović, Sur une fonctionnelle, Publ. Inst. Math. (Beograd) (1932), 146–149.

[18] J. Pečarić and J. Peric, Improvements of the Giaccardi and the Petrovic inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39(1) (2012), 65–75.

[19] A. U. Rehman, M. Mudessir, H. T. Fazal and G. Farid, Petrović’s inequality on coordinates and related results, Cogent Math. 3(1) (2016), 11 pp.

[20] J. Rooin, A. Alikhani and M. S. Moslehian, Operator m-convex functions, Georgian Math. J. 25(1) (2018), 93–107.

[21] G. Toader, On a generalization of the convexity, Mathematica 30(53) (1988), 83-87.

[22] G. Toader, Some generalizations of the convexity, in: I. Marusciac and W. W. Breckner (Eds.), Proceedings of the Colloquium on Approximation and Optimization, University of Cluj-Napoca, 1984.

[23] X. Zhang and W. Jiang, Some properties of log-convex function and applications for the exponential function, Comput. Math. Appl. 63(6) (2012), 1111–1116.

[24] B. Xi, F. Qi and T. Zhang, Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions, ScienceAsia 41(51) (2015), 357–361.