HERMITE-HADAMARD INTEGRAL INEQUALITY FOR $(\alpha,m)$-CONVEX FUNCTIONS

Document Type : Original Paper

Author

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

Abstract

In this paper, after introducing the $m$-convexity by Toader, as an intermediate among the general convexity and star shaped property, we bring Hermite-Hadamard integral inequality on $(\alpha,m)$-convex function in the new form. Previous results about the Hermite-Hadamard inequality for $m$-convex functions are part of the results of our theorems. Illustrated examples of $(\alpha,m)$-convex and $m$-convex functions are also included in the article.

Keywords

Main Subjects


[1] Gh. Amirbostaghi, M. Asadi, M.R. Mardanbeigi, m–Convex Structure on b–Metric Spaces, Filomat 35(14) (2021) 4765-4776.
[2] M. Asadi, Gh. Amirbostaghi, Hermite-Hadamard (Hh) Integral Inequality for m-Convex Functions, Journal of Nonlinear and Convex Analysis 23(8) (2022) 1537-1544.
[3] M. K. Bakula, M. E. Özdemir, J. Pecaric, Hadamard type inequalities for m-convex and (α,m)-convex functions, Pure Appl. Math. 9(4) (2008) Article ID 96.
[4] S. S. Dragomir, On some new inequalities of hermite-hadamard type form-convex functions, Tamkang journal of mathematics 33(1) (2002) 45–55.
[5] S.S. Dragomir, I. Gomm, Some Hermite–Hadamard type inequalities for functions whose exponentials are convex, Stud. Univ. Babe, s–Bolyai, Math. 60 (2015) 527–534.
[6] R. George, S. Radenovic, K. P. Reshma, S.Shukla, “ Rectangular b-metric spaces and contractio principle“, J. Nonlinear Sci. Appl. (2015) 8:10051013.
[7] T. Lara, E. Rosales, J. Sanchez, New Properties ofm-convex functions, International Journal of Mathematical Analysis 9(15) (2015) 735 - 742.
[8] V. G. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, Romania (1993).
[9] S. Özcan, Hermite-Hadamard type inequalities for m–convex and (α,m)–convex function, Journal of Inequalities and Applications 175 (2020).
[10] S. Rashid, F. Safdar, A. O. Akdemir, M. A. Noor, K. I. Noor, Some new fractional integral inequalities forexponentially m–convex functions viaextended generalized Mittag–Leffler function, Journal of Inequalities and Applications (2019) 2019:299.
[11] S. Simić, B. Bin-Mohsin, Some generalizations of the Hermite-Hadamard integral inequality, J. Inequal. Appl. (2021) 2021:72. https://doi.org/10.1186/s13660–021–02605–y
[12] S. Simons, From Hahn-Banach to Monotonicity, Springer, Berlin, (2008).
[13] B. Simon, Convexity: An Analytic Viewpoint, Cambridge University Press, New York, (2011).
[14] G. Toader, The order of starlike convex function, Bull. Appl. Comp. Math. 85 (1998) 347–350.
[15] G. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, Proc. Colloq. Approx. Optim, Cluj–Napoca (Romania) (1985) 329–338.