Autour d'un probl\`eme extr\'emal \'etudi\'e par Edmund Landau

TL;DR

This paper introduces Landau's problem of bounding derivatives based on function bounds, exploring its variants and connections to convex analysis, numerical integration, and approximation theory.

Contribution

It provides an exposition on Landau's derivative bounding problem, highlighting its variants, generalizations, and the structure of extremal functions.

Findings

Characterization of extremal functions

Connections to convex and functional analysis

Insights into derivative bounds and their applications

Abstract

This expository article is an introduction to Landau's problem of bounding the derivative, knowing bounds for the function and its second derivative, and some of its variants and generalizations. Connexions with convex and functional analysis, numerical integration, and approximation theory are emphasized. Among others, we describe the set of extreme points of the relevant set of functions.