On Hopital-style rules for monotonicity and oscillation

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.