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-Fejer type inequality for f-preinvex functions. Moreover, we use these new identities to prove some bounds for the Hermite-Hadamard-Fejer type inequality for generalized conformable K-fractional integrals regarding phi-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.