Chain rules and inequalities for the BHT fractional calculus on arbitrary time scales

TL;DR

This paper extends fractional calculus to arbitrary time scales by establishing chain rules and inequalities, unifying discrete and continuous cases and generalizing classical results.

Contribution

It introduces the Benkhettou-Hassani-Torres fractional calculus on time scales, providing new chain rules and inequalities for fractional derivatives and integrals.

Findings

Derived two chain rules for the fractional derivative.

Proved five inequalities for the fractional integral.

Results unify continuous and discrete fractional calculus.

Abstract

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on time scales by proving two chain rules for the $\alpha$-fractional derivative and five inequalities for the $\alpha$-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order $\alpha = 1$ and the time scale coincides with the set of real numbers.