The Eremenko-Lyubich constant

Abstract

Eremenko and Lyubich proved that an entire function whose set of singular values is bounded is expanding at points where its image has large modulus. These expansion properties have been at the centre of the subsequent study of this class of functions, now called the Eremenko-Lyubich class. We improve the estimate of Eremenko and Lyubich, and show that the new estimate is asymptotically optimal. As a corollary, we obtain an elementary proof that functions in the Eremenko-Lyubich class have lower order at least $1/2$.

5 pages. V2: Author accepted manuscript; minor corrections from V1. To appear in Bull. London Math. Soc

Cited by in corpus (6)

• Eremenko's conjecture, wandering Lakes of Wada, and maverick points
• The escaping set in transcendental dynamics
• A bouquet of pseudo-arcs
• Spiders' webs in the Eremenko-Lyubich class
• Entire functions with Cantor bouquet Julia sets
• Second order linear differential equations with a basis of solutions having only real zeros