Monotonicity of functions and sign changes of their Caputo derivatives

TL;DR

This paper establishes that a function's monotonicity on an interval is equivalent to the non-changing sign of all its Caputo derivatives of order between 0 and 1, extending classical derivative sign criteria.

Contribution

It proves the equivalence between monotonicity and the sign constancy of all Caputo derivatives of order in (0,1), clarifying misconceptions about partial sign constancy.

Findings

Monotonicity is characterized by sign constancy of all Caputo derivatives of order in (0,1).

Partial constancy of Caputo derivatives' signs is insufficient to guarantee monotonicity.

The classical derivative sign criterion extends to fractional derivatives in the Caputo sense.

Abstract

It is well known that a continuously differentiable function is monotone in an interval $[a,b]$ if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders $\alpha \in (0,1)$ with starting point $a$ of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it not sufficient if the Caputo derivatives have a constant sign for a few values of $\alpha \in (0,1)$ only.