On some inequalities involving the function and its fractional derivative

Abstract

This note is concerned with some inequalities involving the function and its Caputo-type Erdélyi-Kober fractional derivative. Namely, we establish integral inequalities of Poincaré, Sobolev, and Hilbert-Pachpatte types, with respect to the right-hand sided Caputo-type modification of the Erdélyi-Kober fractional derivative. The obtained inequalities are useful in analyzing fractional differential equations, in particular, in obtaining various estimate for the solutions.