Generalized convexity of the Lambert $W$ function

TL;DR

This paper explores the generalized convexity and concavity properties of the Lambert W function using H"older means, providing necessary and sufficient conditions and related inequalities.

Contribution

It introduces a comprehensive analysis of Lambert W's $H_{p,q}$-convexity and concavity, characterizing these properties through parameter regions and establishing new inequalities.

Findings

Characterizes strict $H_{p,q}$-convexity and concavity of $W$ on $(0,+ abla)$.

Provides necessary and sufficient conditions based on $(p,q)$ parameters.

Establishes inequalities involving harmonic, geometric, and arithmetic means.

Abstract

This paper investigates the generalized convexity properties of the Lambert $W$ function, defined as the solution to $W(z)e^{W(z)}=z$. Focusing on $H_{p,q}$-convexity and concavity with respect to H\"older means, we derive necessary and sufficient conditions for $W$ to exhibit strict $H_{p,q}$-convexity or concavity on the interval $(0,+\infty)$. The main result characterizes these properties in terms of specific parameter regions $(p,q)$-plane. Inequalities involving harmonic, geometric, and arithmetic means are established, with equalities holding only when $x=y$.