Asymptotic estimates for entire functions of minimal growth with given zeros

Abstract

Let $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we denote the class of all entire functions whose zeros are precisely the $\zeta_n$, where a complex number that occurs $m$ times in the sequence $\zeta$ corresponds to a zero of multiplicity $m$. Suppose that $\Phi$ is a convex function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$ as $\sigma\to+\infty$. It is proved that there exists an entire function $f\in\mathcal{E}_\zeta$ such that$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln n_\zeta( r)}{\Phi(\ln r)},$$where $M_f(r)$ denotes the maximum modulus of the function $f$, and it is shown that the above inequality implies the inequality$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln N_\zeta( r)}{\Phi(\ln r)}+\varlimsup_{\sigma\to+\infty}\frac{\ln\Phi'_+(\sigma)}{\Phi(\sigma)}.$$The formulated result is a consequence of the following more general statement: if the right-hand derivative $\Phi'_+$ of the function $\Phi$ assumes only integer values and $\sum_{n=1}^\infty e^{-\Phi(\ln|\zeta_n|)}<+\infty$, then there exists an entire function $f\in\mathcal{E}_\zeta$ such that $\ln M_f(r)=o(e^{\Phi(\ln r)})$ as $r \to+\infty$.