Growth of Entire Functions with Given Zeros and Representation of Meromorphic Functions

Abstract

Let $\Lambda = \{ \lambda n\}$ be a sequence of points on the complex plane, and let $\Lambda (r)$ be the number of points of the sequence $\Lambda$ in the disk $\{ \left| z \right| < r\}$. We study the following problem in terms of the counting function $\Lambda (r)$: what is the minimal possible growth of the characteristic $M_f (r) = \max \{ \left| {f\left( z \right)} \right|:\left| z \right| = r\}$ in the class of all entire functions $f\not \equiv 0$ vanishing on $\Lambda$? Let $F$ be a meromorphic function in $\mathbb{C}$. In terms of the Nevanlinna characteristic $T_F (r)$ of the function $F$, we estimate the minimal possible growth of the characteristics $M_g (r)$ and $M_h (r)$ in the class of all pairs of entire functions $g$ and $h$ such that $F = g/h$. We present analogs of the obtained results for holomorphic and meromorphic functions in the unit disk in the complex plane.