On zeros and Taylor coefficients of entire function of logarithmic growth

Abstract

In the paper for an important class of entire functions of zero order we find out straightforward relations between the increasing rate of the sequences of zeroes and the decay rate of the Taylor coefficients. Applying the coefficient characterization of the growth of entire functions and some Tauberian theorems from the convex analysis, we obtain asymptotically sharp estimates relating the zeroes $\lambda_n$ and Hadamard rectified Taylor coefficients~$\hat{f_n}$ for entire functions of the logarithmic growth. In the cases, when the function possesses a regular behavior of some kind, the mentioned estimates become asymptotically sharp formulas. For instance, if an entire function has a Borel regular growth and the point $a=0$ is not its Borel exceptional value, then as $n\to\infty$ the asymptotic identity $\ln |\lambda_n|\sim \ln(\hat{f}_{n-1}/\hat{f_n})$ holds true. The result is true for the functions of perfectly regular logarithmic growth and in the latter case we can additionally state that $\ln|\lambda_1\lambda_2\,\ldots\,\lambda_n|\sim\ln\hat{f_n}^{-1}$ as $n\to\infty$.