Abstract
In this work, we introduce new definitions of left and right-sides generalized conformable K-fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for ϕ-preinvex functions. Moreover, we use these new identities to prove some bounds for the Hermite-Hadamard-Fejér type inequality for generalized conformable K-fractional integrals regarding ϕ-preinvex functions. Finally, we also present some applications of the generalized definitions for higher moments of continuous random variables, special means, and solutions of the homogeneous linear Cauchy-Euler and homogeneous linear K-fractional differential equations to show our new approach.
Highlights
The field of fractional calculus is the generalization of classical differential and integral calculus
In [6], Noor et al introduced the concept of φ-invex set and φ-preinvex functions and related all properties, as follows: Definition 1
Fractional calculus is a branch of mathematics that deals with studying and applying arbitrary order integrals and derivatives
Summary
The field of fractional calculus is the generalization of classical differential and integral calculus. In [1], the remarkable inequality is stated as: Let F : I ⊆ < → < be a convex function on the interval I of real numbers and κ1 , κ2 ∈ I with κ1 < κ2. Many researchers have generalized the Hermit-Hadamard inequality using the classical convex function. In [2], Fejér gave a weighted generalization of the inequality (1) for a convex function. In [6], Noor et al introduced the concept of φ-invex set and φ-preinvex functions and related all properties, as follows: Definition 1. A function F on the φ-invex set Ω is said to be φ-preinvex with respect to φ and θ, if. Researchers have expanded their work on Hermite-Hadamard-Fejér type inequalities in the fractional domain by using a wide application of fractional calculus. Hermite-Hadamard-Fejér type inequalities for various classes of function have been identified using fractional integrals
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