On the Periodicity of Entire Functions
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Results in Mathematics
On the Periodicity of Entire Functions Weiran L¨ u and Xiaoxue Zhang Abstract. The purpose of this paper is mainly to prove that if f is a transcendental entire function of hyper-order strictly less than 1 and f (z)n + a1 f (z) + · · · + ak f (k) (z) is a periodic function, then f (z) is also a periodic function, where n, k are positive integers, and a1 , · · · , ak are constants. Meanwhile, we offer a partial answer to Yang’s Conjecture, theses results extend some previous related theorems. Mathematics Subject Classification. 30D35, 39A10. Keywords. Periodicity, entire function, order.
1. Introduction and Main Results Herein let f denote a non-constant meromorphic function and we assume that the reader is familiar with the fundamental results of Nevanlinna theory and its standard notation such as m(r, f ), N (r, f ), T (r, f ), etc (see e.g., [4] and [11]). In the sequel, S(r, f ) will be used to denote a quantity that satisfies S(r, f ) = o T (r, f ) as r → ∞, outside possibly an exceptional set of r values of finite linear measure, and a meromorphic function a is said to be a small function of f if T (r, a) = S(r, f ). We use ρ(f ) and ρ2 (f ) to denote the order and hyper-order of f respectively. The convergence exponent of zeros of f is defined as log N (r, f1 )
log n(r, f1 )
. log r log r r→∞ In addition, a complex number a is said to be a Borel exceptional value of f if 1 log+ n r, f −a lim sup < ρ(f ). log r r→∞ τ (f ) = lim sup r→∞
= lim sup
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W. L¨ u and X. Zhang
Results Math
In this note, we mainly consider the periodicity of entire functions, namely, if f (z)n + a1 f (z) + · · · + ak f (k) (z) is a periodic function, then f (z) is also a periodic function. The motivation of this paper arises from the study of the real transcendental entire solutions of the differential equation f (z)f (k) (z) = p(z) sin2 z, where p(z) is a non-zero polynomial. It seems to us that Titchmarsh [9] firstly proved that the differential equation f (z)f (z) = − sin2 z has no real entire solutions of finite order other than f (z) = ± sin z. The follow-up works were due to Li, L¨ u and Yang in [8], where they considered the similar problem when f (z) is real and of finite order. They obtained f (z)f (z) = − sin2 z has entire solutions f (z) = ± sin z and no other solutions. Recently, Yang proposed the following interesting conjecture, see e.g., [8] and [10]. Yang’s Conjecture. Let f be a transcendental entire function and k (≥ 1) be an integer. If f (z)f (k) (z) is a periodic function, then f (z) is also a periodic function. From then on, a number of papers have focused on Yang’s Conjecture, see e.g., [6,7] and references therein. Recently, regarding Yang’s Conjecture, Liu et al. [5] obtained the following result. Theorem A. Let f be a transcendental entire function and n, k be positive integers. If f (z)n f (k) (z) is a periodic function and one of the following conditions is satisfied (i) k = 1; (ii) f (z) = eh(z) , where h is a non-constant polyn
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