On Hopital-style rules for monotonicity and oscillation

TL;DR

This paper generalizes Hopital-style rules for monotonicity and oscillation, connecting them to properties of convex/concave functions, and extends their applicability beyond differentiability and monotonicity constraints.

Contribution

It introduces generalized Hopital-style rules that do not require differentiability and can handle non-zero initial values and non-monotone derived functions.

Findings

Rules are applicable without differentiability.

Extended to functions with non-zero initial values.

Can handle non-monotone derived functions.

Abstract

We point out the connection of the so-called H\^opital-style rules for monotonicity and oscillation to some well-known properties of concave/convex functions. From this standpoint, we are able to generalize the rules under no differentiability requirements and greatly extend their usability. The improved rules can handle situations in which the functions involved have non-zero initial values and when the derived functions are not necessarily monotone. This perspective is not new; it can be dated back to Hardy, Littlewood and Polya.