The growth of entire functions of genus zero

Dang Duc Trong; Truong Trung Tuyen

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The growth of entire functions of genus zero

Dang Duc Trong·Truong Trung Tuyen

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Abstract

In this paper we shall consider the assymptotic growth of $|P_n(z)|^{1/k_n}$ where $P_n(z)$ is a sequence of entire functions of genus zero. Our results extend a result of J. Muller and A. Yavrian. We shall prove that if the sequence of entire functions has a geometric growth at each point in a set $E$ being non-thin at $\infty$ then it has a geometric growth in $\CC$ also. Moreover, if $E$ has some more properties, a similar result also holds for a more general kind of growth. Even in the case where $P_n$ are polynomials, our results are new in the sense that it does not require $k_n\succeq deg(P_n)$ as usually required.

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Dang Duc Trong, Truong Trung Tuyen. 2006-10-01. The growth of entire functions of genus zero. https://arxiv.org/abs/math/0610045

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