On linear and quadratic Lipschitz bounds for twice continuously differentiable functions

TL;DR

This paper derives explicit piecewise linear and quadratic Lipschitz bounds for twice continuously differentiable functions, based on their Lipschitz constants and those of their derivatives, useful for analysis and applications.

Contribution

It provides explicit, easily applicable bounds for such functions, filling a gap in literature where these are often not readily available.

Findings

Provides explicit Lipschitz bounds for functions and derivatives

Bounds are expressed in terms of Lipschitz constants

Useful for analysis and engineering applications

Abstract

Lower and upper bounds for a given function are important in many mathematical and engineering contexts, where they often serve as a base for both analysis and application. In this short paper, we derive piecewise linear and quadratic bounds that are stated in terms of the Lipschitz constants of the function and the Lipschitz constants of its partial derivatives, and serve to bound the function's evolution over a compact set. While the results follow from basic mathematical principles and are certainly not new, we present them as they are, from our experience, very difficult to find explicitly either in the literature or in most analysis textbooks.