The Eremenko-Lyubich constant

TL;DR

This paper refines the expansion estimates for functions in the Eremenko-Lyubich class, demonstrating their asymptotic optimality and establishing a lower order bound of at least 1/2.

Contribution

It improves the known expansion estimate for the Eremenko-Lyubich class and proves its asymptotic optimality, also providing a new proof of the lower order bound.

Findings

Improved expansion estimate for the Eremenko-Lyubich class

Asymptotic optimality of the new estimate

Functions in the class have lower order at least 1/2

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 $1/2$.