Bank-Laine Functions with Real Zeros
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Bank-Laine Functions with Real Zeros J. K. Langley1 Received: 28 February 2020 / Revised: 18 June 2020 / Accepted: 30 June 2020 © The Author(s) 2020
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) = ±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. Keywords Bank-Laine function · Entire function · Zeros Mathematics Subject Classification 30D20 · 30D35
1 Introduction For a non-constant entire function f , denote by ρ( f ) = lim sup r →+∞
log+ N (r , 1f ) log+ T (r , f ) , λ( f ) = lim sup ≤ ρ( f ), log r log r r →+∞
its order of growth and the exponent of convergence of its zeros [10]. In their seminal paper [1], Bank and Laine proved several landmark results on the oscillation of solutions of y + A(z)y = 0, (1) in which A is an entire function. Their approach was based on taking linearly independent solutions f 1 , f 2 of (1), normalised so as to have Wronskian W ( f 1 , f 2 ) =
Dedicated to the memory of Stephan Ruscheweyh. Communicated by Vladimir V. Andrievskii.
B 1
J. K. Langley [email protected] School of Mathematical Sciences, University of Nottingham, Nottingham, England NG7 2RD, UK
123
J. K. Langley
f 1 f 2 − f 1 f 2 = 1, and then considering the product E = f 1 f 2 , which satisfies 4A =
E E
2 −2
1 E − 2. E E
(2)
In particular, it was shown in [1] that if λ(E) + ρ(A) < +∞ then ρ(E) < +∞, whereas if A is transcendental then the quotient U = f 1 / f 2 always has infinite order, since [14, Ch. 6] U (z) 3 U (z) 2 − SU (z) = = 2 A, (3) U (z) 2 U (z) where SU (z) is the Schwarzian derivative. The following results were proved by Bank and Laine, Rossi and Shen [1,22,23]. Theorem 1.1 ([1,22,23]) Let A be an entire function, let f 1 , f 2 be linearly independent solutions of (1) and let E = f 1 f 2 , so that λ(E) = max{λ( f 1 ), λ( f 2 )}. (i) If A is a polynomial of degree n > 0 then λ(E) = (n + 2)/2. (ii) If λ(E) < ρ(A) < +∞ then ρ(A) ∈ N = {1, 2, . . .}. (iii) If A is transcendental and ρ(A) ≤ 1/2 then λ(E) = +∞, while if 1/2 < ρ(A) < 1 then 1 1 + ≤ 2. ρ(A) λ(E)
(4)
Theorem 1.1(ii) inspired the Bank-Laine conjecture, to the effect that if A is a transcendental entire function and f 1 , f 2 are linearly independent solutions of (1) with λ( f 1 f 2 ) finite then ρ(A) ∈ N ∪ {+∞}. This conjecture has recently been disproved, however, 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). The main thrust of this paper concerns the location of zeros of Bank-Laine functions, these being entire functions E such that E(z) = 0 implies E (z) = ±1. By [2, Lem. C], an entire function E is a Bank-Lain
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