Investigating synchronous function inequalities and exponential kernel-based fractional calculus in population dynamics

Abstract

The present paper is focused on developing Chebyshev-type inequalities using the fractional-order integrals with exponential kernels. We obtain new comparison results for synchronous functions and by the method of mathematical induction we will use these inequalities for a family of non-negative increasing functions. Also we prove a related Chebyshev-type inequality with fractional integral operators under conditions of monotonicity of the functions under consideration. These findings enrich the theory of inequalities in the fractional calculus that supplies methods used in math analysis, engineering, and other areas where memory impacts or time delays exist.