Entire functions of minimal growth with prescribed zeros

Abstract

Let $l$ be a positive continuous increasing to $+\infty$ function on $\mathbb{R}$. For a positive non-decreasing on $\mathbb{R}$ function $h$, we found sufficient and necessary conditions under which, for an arbitrary complex sequence $(\zeta_n)$ such that $\zeta_n\to\infty$ as $n\to\infty$ and $\ln n(r)\ge l(\ln r)$ for all sufficiently large $r$, there exists an entire function $f$ whose zeros are the $\zeta_n$ (with multiplicities taken into account) satisfying $$\ln\ln M(r)=o\big(l^{-1}(\ln n(r))\ln n_{\zeta}(r)h(\ln n(r))\big),\quad r\notin E,\ r\to+\infty,$$ where $E\subset[1,+\infty)$ is a set of finite logarithmic measure. Here, $n(r)$ is the counting function of the sequence $(\zeta_n)$, and $M(r)$ is the maximum modulus of the function $f$.