单位圆盘上亚纯函数的零点和极点的一些注意事项

I. Sheparovych
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Sheparovych","doi":"10.31861/bmj2021.02.10","DOIUrl":null,"url":null,"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.\nUsing 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$N (r,1/f) \\leq A_1 \\eta\\left(\\frac{B_1}{1-r}\\right)$, $N(r,f)\\leq A'_1\\eta \\left( \\frac{B'_1}{1-r}\\right) $,\n$$\\frac1{2k}\\left|\\sum\\limits_{r_1 <|\\lambda_{\\nu}|\\leqslant r_{2}} \\frac1{\\lambda_{\\nu}^k} -\\sum\\limits_{r_1 < |\\mu_j|\\leqslant r_2} \\frac 1{\\mu_j^{k}} \\right| \\leq \\frac{A_{2}}{r_{1}^{k}}\\eta\\left(\\frac{B_{2}}{1 -r_1}\\right ) +\\frac{A_{2}}{r_{2}^{k}}\\max\\left\\{ 1;\\frac 1{k\\ln \\sigma}\\right\\}\\eta\\left(\\frac{B_{2}}{1 -\\sigma r_{2}}\\right)$$\nIt 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\n$$\\frac1{2k}\\left|\\sum\\limits_{r_1 <|\\lambda_{\\nu}|\\leqslant r_{2}} \\frac1{\\lambda_{\\nu}^k} \\right| \\leq \\frac{A_{2}}{r_{1}^{k}}\\eta\\left(\\frac{B_{2}}{1-r_{1}}\\right) +\\frac{A_{2}}{r_{2}^{k}}\\max\\left\\{ 1;\\frac 1{k\\ln \\sigma}\\right\\}\\eta\\left(\\frac{B_{2}}{1 -\\sigma r_{2}}\\right)$$\nthere 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.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"SOME NOTICES ON ZEROS AND POLES OF MEROMORPHIC FUNCTIONS IN A UNIT DISK FROM THE CLASSES DEFINED BY THE ARBITRARY GROWTH MAJORANT\",\"authors\":\"I. 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摘要

在[4]中,用傅里叶系数法得到了在单位磁盘$\{z:|z|0, B>0$上全纯的零序列$(\lambda_{\nu})$和对于$r\in(0;1):\ T(r;f)\leqslant A\eta\left(\frac B{1-|z|}\right)$,其中$T(r;f):=m(r;f)+N(r;f);\ m(r;f)=\frac{1}{2\pi }\int\limits_0^{2\pi } \ln ^{+}|f(re^{i\varphi})|d\varphi$,然后对于某些正常数$A_1, A’_1, B_1, B’_1, A_2, B_2$和对于$k \in\mathbb{N}$, $r,\ r_1$从$(0;1)$, $r_2\in(r_1;1)$和$\sigma\in(1;1/r_2)$的下一个条件为$N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$, $N(r,f)\leq A'_1\eta \left( \frac{B'_1}{1-r}\right) $,$$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} -\sum\limits_{r_1 < |\mu_j|\leqslant r_2} \frac 1{\mu_j^{k}} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1 -r_1}\right ) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$还表明,如果序列$(\lambda_{\nu})$满足条件$N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$和$$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1-r_{1}}\right) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$,则可以从类$T(r;f)\leqslant \frac{A}{\sqrt{1-r}}\eta\left(\frac B{1-r}\right)$构造一个亚纯函数,其中给定的序列是零或极点序列。
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SOME NOTICES ON ZEROS AND POLES OF MEROMORPHIC FUNCTIONS IN A UNIT DISK FROM THE CLASSES DEFINED BY THE ARBITRARY GROWTH MAJORANT
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.
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