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

B. N. Khabibullin

Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 120-134

Résumé

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 $\{|z|$. We study the following problem in terms of the counting function $\Lambda(r)$: what is the minimal possible growth of the characteristic $M_f(r)=\max\{|f(z)|\colon|z|=r\}$ in the class of all entire functions $f\not\equiv0$ 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.