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Real Entire Functions with Real Zeros and a Conjecture of Wiman

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Real Entire Functions with Real Zeros and a Conjecture of Wiman Real entire functions with real zeros and a conjecture of Wiman Walter Bergweiler,∗ A. Eremenko† and J.K. Langley‡ October 24, 2018 1 Introduction An entire function is called real if it maps the real line into itself. The main result of this paper is Theorem 1.1 For every real entire function of infinite order with only real zeros, the second derivative has infinitely many non-real zeros. This conclusion is not true for the first derivative as the example exp(sin z) shows. For real entire functions with finitely many zeros, all of them real, Theorem 1.1 was proved in [3]. Theorem 1.1 can be considered as an exten- sion to functions of infinite order of the following result of Sheil-Small [20], conjectured by Wiman in 1914 [1, 2]. For every integer p ≥ 0, denote by V2p the set of entire functions of the form f(z) = exp(−az2p+2)g(z), arXiv:math/0207229v1 [math.CV] 25 Jul 2002 where a ≥ 0 and g is a real entire function with only real zeros of genus at most 2p + 1, and set U0 = V0 and U2p = V2p\V2p−2 for p ≥ 1. Thus the class of all real entire functions of finite order with real zeros is represented as a union of disjoint subclasses U2p, p =0, 1,.... ∗Supported by the German–Israeli Foundation for Scientific Research and Development, grant GIF G-643-117.6/1999, and by INTAS-99-00089. †Supported by the NSF grant DMS 0100512 and by the Humboldt Foundation. ‡Supported by DAAD 1 ′′ Theorem A (Sheil-Small) If f ∈ U2p then f has at least 2p non-real zeros. Applying Theorem 1.1 to functions of the form z f(z) = exp g(ζ) dζ Z0 we obtain Corollary 1.1 For every real transcendental entire function g, the function g′ + g2 has infinitely many non-real zeros. For polynomials g the corresponding result was conjectured in [4, Probl. 2.64 and 4.28] and proved in [20]: If g is a real polynomial then g′ + g2 has at least deg g − 1 non-real zeros. Corollary 1.1 also follows from the result of Bergweiler and Fuchs [3]. Theorems 1 and A together imply the following Corollary 1.2 If f is a real entire function and ff ′′ has only real zeros then f ∈ U0. We recall that U0, the Laguerre–P´olya class, coincides with the closure of the set of all real polynomials with only real zeros, with respect to uniform convergence on compact subsets of the plane. This was proved by Laguerre [12] for the case of polynomials with positive zeros and by P´olya [18] in the general case. It follows that U0 is closed under differentiation, so that all derivatives of a function f ∈ U0 have only real zeros. P´olya [18] asked if the converse is true: if all derivatives of a real entire function f have only real zeros then f ∈ U0. This conjecture was proved by Hellerstein and Williamson [8, 9]. More precisely, they showed that for a real entire function ′ ′′ f, the condition that ff f has only real zeros implies f ∈ U0. Our Corollary 1.2 shows that in this result one can drop the assumption on the zeros of f ′, as Hellerstein and Williamson conjectured [4, Probl. 2.64]. For the early history of results on the conjectures of Wiman and P´olya we refer to [8, 15], which contain ample bibliography. The main result of Levin and Ostrovskii [15] is Theorem B If f is a real entire function and all zeros of ff ′′ are real then log+ log+ |f(z)| = O(|z| log |z|), z →∞. (1) 2 This shows that a function satisfying the assumptions of Corollary 1.2 cannot grow too fast, but there is a gap between Theorem B and Theorem A. Our Theorem 1.1 bridges this gap. One important tool brought by Levin and Ostrovskii to the subject was a factorization of the logarithmic derivative of a real entire function f with only real zeros: f ′ = ψφ, f where φ is a real entire function, and either ψ is a meromorphic function which maps the upper half-plane H = {z : Im z > 0} into itself or ψ ≡ 1. This factorization was used in all subsequent work in the subject. A standard estimate for analytic functions mapping the upper half-plane into itself shows that ψ is neither too large nor too small away from the real axis, so the asymptotic behavior of f ′/f mostly depends on that of φ. One can show that f is of finite order if and only if φ is a polynomial. The second major contribution of Levin and Ostrovskii was the appli- cation of ideas from the value distribution theory of meromorphic functions [5, 7, 16]. Using Nevanlinna theory, Hayman [6] proved that for an entire function f, the condition f(z)f ′′(z) =06 , z ∈ C, implies that f ′/f is constant. The assumptions of Theorem B mean that f(z)f ′′(z) =6 0 in H. Levin and Os- trovskii adapted Hayman’s argument to functions in a half-plane to produce an estimate for the logarithmic derivative. An integration of this estimate gives (1). To estimate the logarithmic derivative using Hayman’s argument they applied an analogue of the Nevanlinna characteristic for meromorphic functions in a half-plane, and proved an analogue of the main technical result of Nevanlinna theory, the lemma on the logarithmic derivative. This char- acteristic has two independent origins, [13] and [21], and the name “Tsuji characteristic” was introduced in [15]. In this paper we use both main ingredients of the work of Levin and Ostrovskii, the factorization of f ′/f and the Tsuji characteristic. Another important tool comes from Sheil-Small’s proof of Theorem A. His key idea was the study of topological properties of the auxiliary function f(z) F (z)= z − . f ′(z) In the last section of his paper, Sheil-Small discusses the possibility of exten- sion of his method to functions of infinite order, and proves the fact which 3 turns out to be crucial: if f is a real entire function, ff ′′ has only real zeros, and f ′ has a non-real zero, then F has a non-real asymptotic value. In §4 we prove a generalization of this fact needed in our argument. The auxiliary function F appears when one solves the equation f(z)=0 by Newton’s method. This suggests the idea of iterating F and using the Fatou–Julia theory of iteration of meromorphic functions. This was explored by Eremenko and Hinkkanen, see, for example, [10]. Theorem 1.1 will be proved by establishing a more general result con- jectured by Sheil-Small [20]. Let L be a real meromorphic function in the plane with only simple poles, all of them real and with positive residues. It is known [8, 15, 20] that every such L has a Levin–Ostrovskii representation L = ψφ (2) in which: (a) ψ is meromorphic in the plane and real on the real axis; (b) ψ maps the upper half-plane into itself, or ψ ≡ 1; (c) ψ has a simple pole at every pole of L, and no other poles; (d) φ is a real entire function. We outline briefly how such a factorization (2) is obtained. Let ...<ak−1 < ak < ak+1 < . be the sequence of poles of L enumerated in increasing order. The assumption that all poles are simple and have positive residues implies that there is at least one zero of L in each interval (ak, ak+1). We choose one such zero in each interval and denote it by bk. Then we set 1 − z/b ψ(z)= k , 1 − z/ak Yk with slight modifications if akbk ≤ 0 for some k or the set {ak} is bounded above. We then define φ by (2), and properties (a)–(d) follow (for the details see [8, 15, 20]). Theorem 1.2 Let L be a function meromorphic in the plane, real on the real axis, such that all poles of L are real, simple and have positive residues. Let ψ,φ be as in (2) and (a), (b), (c), (d). If φ is transcendental then L + L′/L has infinitely many non-real zeros. 4 To deduce Theorem 1.1 from Theorem 1.2 it suffices to note that if f is a real entire function with only real zeros then L = f ′/f is real meromorphic with only real simple poles and positive residues, and thus has a represen- tation (2). Further, L + L′/L = f ′′/f ′. By an argument of Hellerstein and Williamson [9, pp. 500-501], the function φ is transcendental if and only if f has infinite order. 2 Preliminaries We will require the following well known consequence of Carleman’s estimate for harmonic measure. Lemma 2.1 Let u be a non-constant continuous subharmonic function in the plane. For r > 0 let B(r, u) = max{u(z): |z| = r}, and let θ(r) be the angular measure of that subset of the circle C(0,r) = {z ∈ C : |z| = r} on which u(z) > 0. Define θ∗(r) by θ∗(r) = θ(r), except that θ∗(r) = ∞ if u(z) > 0 on the whole circle C(0,r). Then if r > 2r0 and B(r0,u) > 1 we have r + iθ πdt log ku (4re )k ≥ log B(2r, u) − c1 ≥ ∗ − c2, Z2r0 tθ (t) in which c1 and c2 are absolute constants, and 1 π ku+(reiθ)k = max{u(reiθ), 0} dθ.
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