INEQUALITIES VIA STRONGLY p; h -HARMONIC CONVEX FUNCTIONS

Year 2020, Volume: 10 Issue: 1, 81 - 94, 01.01.2020

M. A. Noor K. I. Noor S. Iftikhar

Abstract

The main aim of this paper is to consider a new class of harmonic convex functions with respect to an arbitrary non-negative function, which is called strongly p; h -harmonic convex function. We establish Hermite-Hadamard like integral inequal- ities via these new classes of convex functions. Some special cases are discussed, which can be obtained from our main results. The ideas and techniques of this paper may stimulate further research.

Keywords

Harmonic convex functions, p-convex functions, Hermite-Hadamard type in-equalities.

References

• Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., (2007), Generalized convexity and in- equalities, J. Math. Anal. Appl., 335, pp. 1294-1308.
• Angulo, H., Gimenez, J., Moros, A. M. and Nikodem, K., (2011), On strongly h-convex functions, Ann. Funct. Anal., 2, pp. 87-93.
• Cristescu G. and Lupsa, L., (2002), Non-connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, Holland.
• Hermite, C., (1883), Sur deux limites d’une intgrale dfinie. Mathesis, 3, 82.
• Hadamard, J., (1893), Etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par Riemann. J. Math. Pure Appl., 58, pp. 171-215.
• Iscan, I., (2014), Hermite-Hadamard type inequalities for harmonically convex functions. Hacett, J. Math. Stats., 43(6), pp. 935-942.
• Merentes, N. and Nikodem, K., (2010), Remarks on strongly convex functions, Aequationes Math. 80(1-2), pp. 193-199.
• Nikodem, K. and Pales, Z., (2011), Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. 5(1), pp. 8387.
• Niculescu C. P. and Persson, L. E., (2006), Convex Functions and Their Applications, Springer-Verlag, New York.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2015), Nonconvex functions and integral inequalities, Punjab. Uni. J. Math., 47(2), pp. 19-27.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2016), Hermite-Hadamard inequalities for harmonic non- convex functions, MAGNT Research Report, 4(1), pp. 24-40.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2017), Integral inequalities for differentiable p-harmonic convex functions, Filomat, 31(20), pp. 6575-6584.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2016),Integral inequalities for differentiable relative har- monic preinvex functions(survey), TWMS J. Pure Appl. Math., 7(1), pp. 3-19.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2016), Hermite-Hadamard inequalities for strongly harmonic convex functions, J. Inequ. Special Func., 7(3), pp. 99-113.
• Noor, M. A., Noor, K. I., Iftikhar, S. and Awan, M.U., (2016), Strongly generalized harmonic convex functions and integral inequalities, J. Math. Anal., 7(3), pp. 66-77.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2017), Inequalities via strongly p-harmonic log-convex functions, J. Nonlinear Func. Anal., 2017, Article ID 20.
• Noor, M. A., Noor, K. I. and Iftikhar, S., (2016), Integral inequalities for extended harmonic convex functions, Advanced Math. Models and Appl. 2(3), pp. 216-229.
• Pecaric, J., Proschan, F. and Tong, Y. L., (1992), Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New york.
• Polyak, B.T., (1966), Existence theorems and convergence of minimizing sequences in extremum problems with restictions, Soviet Math. Dokl. 7, pp. 7275.
• Varosanec, S., (2007), On h-convexity, J. Math. Anal. Appl., 326(1), pp. 303-311.
• Vial, J. P., (1982), Strong convexity of sets and functions, J. Math. Economy, 9, pp. 187205.