Improvement of conformable fractional Hermite-Hadamard type inequality for convex functions and some new conformable fractional midpoint type inequalities

Sercan Turhan; ˙Imdat ˙Is¸can; Mehmet Kunt

doi:10.26637/MJM0803/0004

Abstract

In this paper, it is proved that conformable fractional Hermite-Hadamard inequality and conformable fractional Hermite-Hadamard-Fejer inequality is just the results of Hermite-Hadamard-Fejér inequality. After this, a new conformable fractional Hermite-Hadamard inequality which is not a result of Hermite-Hadamard-Fejér inequality and better than given in [8] by Set et al. is obtained. Also, new equality is proved and some new conformable fractional midpoint type inequalities are given. Our results have some relations with the results given in $[5,6]$.

Keywords:

Convex functions, Hermite-Hadamard inequalities, Conformable fractional integrals, Midpoint type inequalities

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Sercan Turhan Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200, Giresun, Turkey. https://orcid.org/0000-0002-4392-2182
˙Imdat ˙Is¸can Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200, Giresun, Turkey. https://orcid.org/0000-0001-6749-0591
Mehmet Kunt Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey. https://orcid.org/0000-0002-8730-5370

Pages: 753-760
Date Published: 01-07-2020
Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)

T. Abdeljawad, On conformable fractional calculus, $J$. Comp. Appl. Math., 279 (2015), 57-66.

L. Fejér, Uber die Fourierreihen, II,Math. Naturwise. Anz Ungar. Akad. Wiss, 24 (1906), 369-390, (in Hungarian).

J. Hadamard, Étude sur les propriétés des fonctionsentières et en particulier d'unefonctionconsidérée par Riemann,J. Math. Pures Appl., 58 (1893), 171-215.

Ch. Hermite, Sur deux limitesd'uneintégraledéfinie, Mathesis, 3 (1883), 82-83.

U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004) $137-146$.

M. Kunt, İ. İşcan, S. Turhan, D. Karapınar, Improvement of fractional Hermite-Hadamard type inequality and some new fractional midpoint type inequalities for convex functions,Miskolc Mathematical Notes, 19(2) (2018), $1007-1017$.

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations.Elsevier, Amsterdam (2006).

E. Set., A. O. Akdemir, İ. Mumcu, Hermite-Hadamard's inequality and its extensions for conformable fractional integrals of any order $alpha>0$, Creat. Math. Inf, 27, 197206.

E. Set, İ. Mumcu, Hermite-Hadamard-Fejer type inequalities for conformable fractional integrals,Miskolc Mathematical Notes, 20(1) (2019), 475-488.

S. Turhan, The left conformable fractional HermiteHadamard type inequalities for convex functions, Filomat, 33(8), 2417-2430.

S. Turhan, İ. İşcan, M. Kunt, The right conformable fractional Hermite-Hadamard type inequalities for convex functions,Available online from: https://www.researchgate.net/publication/318338580,

Y. Zhou, Basic theory of fractional differential equations, World Scientific, New Jersey (2014).