Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

Abstract

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s, m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.

Keywords:

• m-convex functions
• (s, m)-convex functions
• Hermite–Hadamard-type inequalities
• Riemann–Liouville fractional integrals

AMS Subject Classifications::

• 26A33
• 26A51
• 26D15

Acknowledgements

The first and the second authors acknowledge the support by NNSF of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169); the third author acknowledges the support by Grants VEGA-MS 1-0507-11, VEGA-SAV 2-0124-12 and APVV-0414-07; and the fourth author acknowledges the support by NNSF of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).