Bank-Laine Functions with Real Zeros

J K Langley

doi:10.1007/s40315-020-00342-9

Abstract

Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that E'(z) = pm 1 whenever E(z) = 0. Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

Highlights

IntroductionTheorem 1.1(ii) inspired the Bank-Laine conjecture, to the effect that if A is a transcendental entire function and f1, f2 are linearly independent solutions of (1) with λ( f1 f2) finite ρ( A) ∈ N ∪ {+∞}
For a non-constant entire function f, denote by ρ( f ) = lim sup log+ T (r, f ), λ( f ) = lim sup log+ N (r, 1 f ≤ ρ(), r→+∞ log r r →+∞log r its order of growth and the exponent of convergence of its zeros [10]
Theorem 1.1(ii) inspired the Bank-Laine conjecture, to the effect that if A is a transcendental entire function and f1, f2 are linearly independent solutions of (1) with λ( f1 f2) finite ρ( A) ∈ N ∪ {+∞}. This conjecture has recently been disproved, in the first of two remarkable papers of Bergweiler and Eremenko [4,5] which use quasiconformal constructions; in the second of these they show that equality is possible in (4), for every choice of ρ(A) ∈ (1/2, 1)

Summary

Introduction

Theorem 1.1(ii) inspired the Bank-Laine conjecture, to the effect that if A is a transcendental entire function and f1, f2 are linearly independent solutions of (1) with λ( f1 f2) finite ρ( A) ∈ N ∪ {+∞}. This conjecture has recently been disproved, in the first of two remarkable papers of Bergweiler and Eremenko [4,5] which use quasiconformal constructions; in the second of these they show that equality is possible in (4), for every choice of ρ(A) ∈ (1/2, 1).

Preliminaries

A Real Bank-Laine Function with Real Zeros