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A question of Gol’dberg concerning entire functions with prescribed zeros

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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.

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Author notes

  1. Walter Bergweiler

    Present address: Lehrstuhl II für Mathematik, RWTH Aachen, Templergraben 55, W-5100, Aachen, Germany

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  1. Department of Mathematics, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong

    Walter Bergweiler

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  1. Walter Bergweiler
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Bergweiler, W. A question of Gol’dberg concerning entire functions with prescribed zeros. J. Anal. Math. 63, 121–129 (1994). https://doi.org/10.1007/BF03008421

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  • DOI: https://doi.org/10.1007/BF03008421

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