{"title":"Bounds for the derivative of certain meromorphic functions and on meromorphic Bloch-type functions","authors":"Bappaditya Bhowmik, Sambhunath Sen","doi":"10.21136/cmj.2024.0332-23","DOIUrl":null,"url":null,"abstract":"<p>It is known that if f is holomorphic in the open unit disc <span>\\(\\mathbb{D}\\)</span> of the complex plane and if, for some <i>c</i> > 0, ∣<i>f</i>(<i>z</i>)∣ ⩽ 1/(1−∣<i>z</i>∣<sup>2</sup>)<sup><i>c</i></sup>, <span>\\(z \\in \\mathbb{D}\\)</span>, then ∣<i>f</i>′(<i>z</i>)∣ ⩽ 2(<i>c</i>+1)/(1−∣<i>z</i>∣<sup>2</sup>)<sup><i>c</i>+1</sup>. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in D. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/cmj.2024.0332-23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
It is known that if f is holomorphic in the open unit disc \(\mathbb{D}\) of the complex plane and if, for some c > 0, ∣f(z)∣ ⩽ 1/(1−∣z∣2)c, \(z \in \mathbb{D}\), then ∣f′(z)∣ ⩽ 2(c+1)/(1−∣z∣2)c+1. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in D. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.
众所周知,如果 f 在复平面的开放单位圆盘 (\mathbb{D}\)中是全态的,并且对于某些 c >;0,∣f(z)∣ ⩽ 1/(1-∣z∣2)c, \(z \in \mathbb{D}\),那么∣f′(z)∣ ⩽ 2(c+1)/(1-∣z∣2)c+1。我们考虑了这一结果的分形类似物。此外,我们还引入并研究了一类在 D 中具有非零简单极点的布洛赫类函数。