Explore Ghayasudin Fractional Operators and Improve Some Inequalities through Convexity

Abstract

Special functions play a crucial role in the progression from classical calculus to frac- tional calculus, with extensive applications across all fields of pure and applied mathe- matics. In this study, we discuss the different versions of Bessel-Maitland functions and its generalizations, fractional operators having multi-index Bessel-Maitland functions as its kernel. The modification of well-known inequalities like as Hermite-Hadamard (HH), Fejér type inequalities with refinements are also considered by implementation of generalized fractional operators named as Ghayasuddin fractional. We also exam- ine the interrelationships among these well-known inequalities (HH, TI, FI, and MPI). Furthermore, we extend these integral inequalities to the class of (η1, η2)-convex func- tions and derive additional related inequalities as consequences of our main results. Moreover, we discuss the fractional and differential behavior of generalized version of Bessel-Maitland function of six papramenters. 

Keywords: Fractional calculus, Bessel–Maitland functions, Generalized fractional operators, Ghayasuddin fractional operators, Hermite–Hadamard inequalities, Fejér type inequalities