Prof. Z. Tomovski
(University of Skopje)

Title:
Complete monotonicity of the Mittag-Leffler functions and Opial type inequalities for fractional differential and integral operators involving Mittag-Leffler functions 

Abstract:
Completely monotonic functions have applications in different areas of mathematics, for instance, in potential theory, probability theory, numerical and asymptotic analysis and combinatorics, modeling by anomalous diffusion, etc. In this talk we'll present some recent new results on complete monotonicity of three parameter Mittag-Leffler functions, defined by Prabhakar. In 1960, Opial established an integral inequality, which is a fundamental result in the theory of differential or difference equations and other areas of mathematics. Opial-type integral inequalities were considered for different kinds of fractional derivatives and fractional integral operators for example Riemann-Liouville, Caputo, Canavati, etc. We'll presents a class of very general weighted Opial type integral inequalities using integral and differential operators with kernels in fractional calculus involving generalized Mittag-Leffler functions. Namely, ineteresting Opial type inequalities will be given for Hilfer, Prabhakar, Hilfer-Prabhakar, Caputo-Prabhakar and other differential and integral operators in fractional calculus.