On Generalized $\psi $-Conformable Calculus: Properties and Inequalities

Abstract

In this paper, we first introduce a new fractional derivatives and integralscalled generalized $\psi $-conformable derivative and generalized $\psi $%-conformable integral operators, respectively. We also show that theseoperators generalize various well-known fractional integral operators. Then,we present several properties of these operators including semi-groupproperty. Moreover, we apply these operators to obtain a newHermite-Hadamard-type inequality for convex functions. Furthermore, we obtain corresponding midpoint and trapezoid type inequalities for functions whose derivatives in absolute value are convex.