Generalized Convex Function and Associated Petrovic's Inequality

Rehman, A. Ur.; Farid, G.; Mishra, Vishnu Narayan

doi:10.28924/2291-8639-17-2019-122

Research Article

Generalized Convex Function and Associated Petrovic's Inequality

A. Ur. Rehman
G. Farid
Vishnu Narayan Mishra¹*

Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India.

* Corresponding author

Abstract

In this paper, Petrovi ´c's inequality is generalized for h-convex functions, when h is supermultiplicative function. It is noted that the case for h-convex functions does not lead the particular cases for P -function, Godunova-Levin functions, s-Godunova-Levin functions and s-convex functions due to the conditions imposed on h. To cover the case, when h is submultiplicative, Petrovi ´c's inequality is generalized for h-concave functions.

How to Cite

A. U. Rehman, G. Farid, V. N. Mishra, Generalized Convex Function and Associated Petrovic's Inequality, Int. J. Anal. Appl. 17 (2019), 122–131. https://doi.org/10.28924/2291-8639-17-2019-122

BibTeX

@article{Rehman2019,
  author  = {Rehman, A. Ur. and Farid, G. and Mishra, Vishnu Narayan},
  title   = {Generalized Convex Function and Associated Petrovic's Inequality},
  journal = {Int. J. Anal. Appl.},
  volume  = {17},
  year    = {2019},
  pages   = {122--131},
  doi     = {10.28924/2291-8639-17-2019-122}
}

References

[1] S. Varoˇsanec, On h-Convexity, J. Math. Anal. Appl. 326(1) (2007), 303-311.
[2] S. S. Dragomir, J. E. Peˇcari ´c and L. E. Persson , Some Inequalities of Hadamard inequality, Soochow J. Math. 21(3) (1995), 335-341.
[3] A. H ´azy, Bernstein-Doetsch Type Results for h-Convex Functions, Math. Inequal. Appl. 14(3) (2011), 499-508.
[4] 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.
[5] M. Alomari, M. A. Latif, On Hadmard-Type Inequalities for h-Convex Functions on the Co-ordinates, Int. J. Math. Anal. 3(33) (2009), 1645-1656.
[6] M. Petrovi ´c's, Sur Une Fonctionnelle, Publ. Math. Univ. Belgrade 1 (1932), 146-149.
[7] J. E. Peˇcari ´c, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1991.
[8] M. Bombardelli, S. Varoanec, Properties of h-Convex Functions Related to the HermiteHadamardFejr Inequalities, Comput. Math. Appl. 58(9) (2009), 1869-1877.
[9] A. Olbry ´s, On Separation by h-Convex Function, Tatra Mt. Math. Publ. 62(1) (2015), 105-111.
[10] S. Butt, J. Peˇcari ´c and A. U. Rehman, Exponential Convexity of Petrovi ´c and Related Functional, J. Inequal. Appl. 2011(1) (2011), 16 pp.
[11] J. Peˇcari ´c and V. Culjak, Inequality of Petrovi ´c and Giaccardi for Convex Function of Higher Order, Southeast Asian Bull. ˇ Math. 26(1) (2003), 57-61.
[12] J. E. Peˇcari ´c, On the Petrovi ´c Inequality for Convex Functions, Glasnik Matematicki 18(38) (1983), 77-85.
[13] M. K. Bakula, J. Peˇcari ´c and M. Ribiˇci ´c, Companion Inequalities to Jensen's Inequality for m-Convex and (α, m)-Convex Functions, J. Inequal. Pure Appl. Math. 7(5) (2006), 32 pp.
[14] A. U. Rehman, M. Mudessir, H. T. Fazal and G. Farid, Petrovi ´c's Inequality on Coordinates and Related Results, Cogent Math. 3(1) (2016), 11 pp.
[15] J. Peˇcari ´c and J. Peric, Improvements of the Giaccardi and the Petrovic Inequality and Related Stolarsky Type Means, An. Univ. Craiova, Ser. Mat. Inf. 39(1) (2012), 65-75.