A question of Gol’dberg concerning entire functions with prescribed zeros |
| |
Authors: | Walter Bergweiler |
| |
Institution: | 1. Department of Mathematics, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong
|
| |
Abstract: | Let (zj) be a sequence of complex numbers satisfying |zj|→ ∞ asj → ∞ and denote by n(r) the number of zj satisfying |zj|≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫∞ (ϕ(t)t logt)−1
dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here.
These results answer a question by A. A. Gol’dberg. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|