Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions

In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, $ k $-Riemann-Liouville ($ k $-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.

[1]  S. Dragomir,et al.  Hermite–Hadamard type inequalities for conformable fractional integrals , 2017, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas.

[2]  Mujahid Abbas,et al.  Simpson's and Newton's type quantum integral inequalities for preinvex functions , 2021 .

[3]  Thabet Abdeljawad,et al.  Modification of certain fractional integral inequalities for convex functions , 2020 .

[4]  Gerhard Schmeisser,et al.  Sharp Error Estimates for Interpolatory Approximation on Convex Polytopes , 2005, SIAM J. Numer. Anal..

[5]  Dumitru Baleanu,et al.  On the Generalized Hermite-Hadamard Inequalities via the Tempered Fractional Integrals , 2020, Symmetry.

[6]  Pshtiwan Othman Mohammed,et al.  A New Version of the Hermite-Hadamard Inequality for Riemann-Liouville Fractional Integrals , 2020, Symmetry.

[7]  Mehmet Zeki Sarikaya,et al.  On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals , 2017 .

[8]  A. M. Fink,et al.  Jensen inequalities for functions with higher monotonicities , 1990 .

[9]  Pshtiwan Othman Mohammed,et al.  Hermite‐Hadamard inequalities for Riemann‐Liouville fractional integrals of a convex function with respect to a monotone function , 2019, Mathematical Methods in the Applied Sciences.

[10]  Y. Chu,et al.  Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables , 2021 .

[11]  Charles E. M. Pearce,et al.  Selected Topics on Hermite-Hadamard Inequalities and Applications , 2003 .

[12]  Inequalities of trapezoidal type involving generalized fractional integrals , 2020 .

[13]  M. Sarikaya,et al.  SOME NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR s-CONVEX FUNCTIONS , 2015 .

[14]  Ugur S. Kirmaci,et al.  Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula , 2004, Appl. Math. Comput..

[15]  Thabet Abdeljawad,et al.  On conformable fractional calculus , 2015, J. Comput. Appl. Math..

[16]  J. Pečarić,et al.  A variant of Jensen’s inequality of Mercer’s type for operators with applications , 2006 .

[17]  M. Abbas,et al.  Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second q b $q^{b}$ -derivatives , 2021 .

[18]  Mehmet Zeki Sarikaya,et al.  Hermite–Hadamard type inequalities for F-convex function involving fractional integrals , 2018, Journal of inequalities and applications.

[19]  A. Guessab DIRECT AND CONVERSE RESULTS FOR GENERALIZED MULTIVARIATE JENSEN-TYPE INEQUALITIES , 2013 .

[20]  Gerhard Schmeisser,et al.  Sharp Integral Inequalities of the Hermite-Hadamard Type , 2002, J. Approx. Theory.

[21]  Hatice Öğulmüş,et al.  Hermite-Hadamard-Mercer type inequalities for fractional integrals , 2021, Filomat.

[22]  M. Ali,et al.  Simpson and Newton type inequalities for convex functions via newly defined quantum integrals , 2020, Mathematical Methods in the Applied Sciences.

[23]  Ghulam Farid,et al.  ON HADAMARD INEQUALITIES FOR k-FRACTIONAL INTEGRALS , 2016 .

[24]  M. Abbas,et al.  Simpson's and Newton's type Quantum integral inequalities for preinvex functions , 2020 .

[25]  A. Karaca,et al.  On the k-Riemann-Liouville fractional integral and applications , 2014 .

[26]  J. Hadamard,et al.  Etude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann , 1893 .

[27]  Gerhard Schmeisser,et al.  Convexity results and sharp error estimates in approximate multivariate integration , 2003, Math. Comput..

[28]  Zhiyue Zhang,et al.  Some new Simpson's type inequalities for coordinated convex functions in quantum calculus , 2020, Mathematical Methods in the Applied Sciences.

[29]  Hüseyin Budak,et al.  Some New Quantum Hermite–Hadamard-Like Inequalities for Coordinated Convex Functions , 2020, Journal of Optimization Theory and Applications.

[30]  M. Emin Özdemir,et al.  On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula , 2004, Appl. Math. Comput..

[31]  Pshtiwan Othman Mohammed,et al.  On generalized fractional integral inequalities for twice differentiable convex functions , 2020, J. Comput. Appl. Math..

[32]  P. Mohammed,et al.  Some new Hermite-Hadamard type inequalities for MT-convex functions on differentiable coordinates , 2017 .

[33]  M. Abbas,et al.  New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions , 2021 .

[34]  Jen-Chih Yao,et al.  Generalized fractional integral inequalities of Hermite–Hadamard type for (α,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{do , 2019, Journal of Inequalities and Applications.

[35]  Zhiyue Zhang,et al.  On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions , 2021 .

[36]  Mehmet Zeki Sarikaya,et al.  Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities , 2013, Math. Comput. Model..

[37]  Thabet Abdeljawad,et al.  Fractional operators with exponential kernels and a Lyapunov type inequality , 2017, Advances in Difference Equations.

[38]  Zhiyue Zhang,et al.  Some new Simpson's type inequalities for coordinated convex functions in quantum calculus , 2020, Mathematical Methods in the Applied Sciences.

[39]  Ravi P. Agarwal,et al.  Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula , 1998 .

[40]  Jen-Chih Yao,et al.  Generalized fractional integral inequalities of Hermite–Hadamard type for (α,m)${(\alpha,m)}$-convex functions , 2019, Journal of Inequalities and Applications.

[41]  Yu‐ming Chu,et al.  Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus , 2021, Open Mathematics.

[42]  Fatma Ertuğral,et al.  On the generalized Hermite-Hadamard inequalities , 2020 .

[43]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[44]  Arran Fernandez,et al.  Hermite‐Hadamard inequalities in fractional calculus defined using Mittag‐Leffler kernels , 2020, Mathematical Methods in the Applied Sciences.

[45]  Ahmed Alsaedi,et al.  Hermite-Hadamard, Hermite-Hadamard-Fejér, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals , 2016, J. Comput. Appl. Math..

[46]  F. Mainardi Fractional Calculus , 2018, Fractional Calculus.

[47]  M. Moslehian,et al.  Refinements of the operator Jensen-Mercer inequality , 2013 .