SOME NOTICES ON ZEROS AND POLES OF MEROMORPHIC FUNCTIONS IN A UNIT DISK FROM THE CLASSES DEFINED BY THE ARBITRARY GROWTH MAJORANT

Abstract

In [4] by the Fourier coefficients method there were obtained some necessary and sufficient conditions for the sequence of zeros $(\lambda_{\nu})$ of holomorphic in the unit disk $\{z:|z|<1\}$ functions $f$ from the class that determined by the majorant $\eta :[0;+\infty)\to [0;+\infty )$ that is an increasing function of arbitrary growth. Using that result in present paper it is proved that if $(\lambda_{\nu})$ is a sequence of zeros and $(\mu_ {j})$ is a sequence of poles of the meromorphic function $f$ in the unit disk, such that for some $A>0, B>0$ and for all $r\in(0;1):\ T(r;f)\leqslant A\eta\left(\frac B{1-|z|}\right)$, where $T(r;f):=m(r;f)+N(r;f);\ m(r;f)=\frac{1}{2\pi }\int\limits_0^{2\pi } \ln ^{+}|f(re^{i\varphi})|d\varphi$, then for some positive constants $A_1, A’_1, B_1, B’_1, A_2, B_2$ and for all $k \in\mathbb{N}$, $r,\ r_1$ from $(0;1)$, $r_2\in(r_1;1)$ and $\sigma\in(1;1/r_2)$ the next conditions hold $N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$, $N(r,f)\leq A'_1\eta \left( \frac{B'_1}{1-r}\right) $, $$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} -\sum\limits_{r_1 < |\mu_j|\leqslant r_2} \frac 1{\mu_j^{k}} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1 -r_1}\right ) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$ It is also shown that if sequence $(\lambda_{\nu})$ satisfies the condition $N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$ and $$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1-r_{1}}\right) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$ there is possible to construct a meromorphic function from the class $T(r;f)\leqslant \frac{A}{\sqrt{1-r}}\eta\left(\frac B{1-r}\right)$, for which the given sequence is a sequence of zeros or poles.