{"title":"单位圆盘上亚纯函数的零点和极点的一些注意事项","authors":"I. 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. 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\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bukovinian Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31861/bmj2021.02.10\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bukovinian Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31861/bmj2021.02.10","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.