Bohr phenomenon for certain close-to-convex analytic functions
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910126386970624 |
|---|---|
| author | Allu, Vasudevarao Halder, Himadri |
| author_facet | Allu, Vasudevarao Halder, Himadri |
| contents | We say that a class $\mathcal{B}$ of analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_{f}<1$, the following inequality $$ \sum\limits_{n=1}^{\infty} |a_{n}z^{n}| \leq d(f(0),\partial f(\mathbb{D}) ) $$ holds for $|z|=r\leq R_{f}$ and for all functions $f \in \mathcal{B}$. The largest radius $R_{f}$ is called Bohr radius for the class $\mathcal{B}$. In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes $\mathcal{S}_{c}^{*}(ϕ),\,\mathcal{C}_{c}(ϕ),\, \mathcal{C}_{s}^{*}(ϕ),\, \mathcal{K}_{s}(ϕ)$. Using Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we obtain some radius $R_{f}$ such that Bohr phenomenon for these classes holds for $|z|=r\leq R_{f}$. Generally, in this case $R_{f}$ need not be sharp, but we show that under some additional conditions on $ϕ$, the radius $R_{f}$ becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_00187 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Bohr phenomenon for certain close-to-convex analytic functions Allu, Vasudevarao Halder, Himadri Complex Variables 30C45, 30C50, 30C80 We say that a class $\mathcal{B}$ of analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_{f}<1$, the following inequality $$ \sum\limits_{n=1}^{\infty} |a_{n}z^{n}| \leq d(f(0),\partial f(\mathbb{D}) ) $$ holds for $|z|=r\leq R_{f}$ and for all functions $f \in \mathcal{B}$. The largest radius $R_{f}$ is called Bohr radius for the class $\mathcal{B}$. In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes $\mathcal{S}_{c}^{*}(ϕ),\,\mathcal{C}_{c}(ϕ),\, \mathcal{C}_{s}^{*}(ϕ),\, \mathcal{K}_{s}(ϕ)$. Using Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we obtain some radius $R_{f}$ such that Bohr phenomenon for these classes holds for $|z|=r\leq R_{f}$. Generally, in this case $R_{f}$ need not be sharp, but we show that under some additional conditions on $ϕ$, the radius $R_{f}$ becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes. |
| title | Bohr phenomenon for certain close-to-convex analytic functions |
| topic | Complex Variables 30C45, 30C50, 30C80 |
| url | https://arxiv.org/abs/2008.00187 |