Generalized convexity of the Lambert W function

2025/08/24 by Wang, Gendi
#26D07 #33E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.17409

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 Hölder means, we derive necessary and sufficient conditions for W to exhibit strict H_p,q-convexity or concavity on the interval (0,+∞). 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.

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