The growth of entire functions of genus zero

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 $k_n⪰deg(P_n)$ as usually required.

13 pages

Cited by in corpus (1)

• Sets non-thin at \infty in \Bbb C ^m, and the growth of sequences of entire functions of genus zero