In this paper, we first determine Bohr's inequality for the class of harmonic mappings $f=h+\overline{g}$ in the unit disk $\ID$, where either both $h(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ and $g(z)=\sum_{n=0}^{\infty}b_{pn+m}z^{pn+m}$ are analytic and bounded in $\ID$, or satisfies the condition $|g'(z)|\leq d|h'(z)|$ in $\ID\backslash \{0\}$ for some $d\in [0,1]$ and $h$ is bounded. In particular, we obtain Bohr's inequality for the class of harmonic $p$-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp.
In this article, we determine sharp Bohr-type radii for certain complex integral operators defined on a set of bounded analytic functions in the unit disk.
In this paper, we generalize and investigate Bohr-Rogosinski's inequalities and the Bohr-Rogosinski phenomenon for the subfamilies of univalent (i.e., one-to-one) functions defined on unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1 \}$ which maps to the concave domain, i.e., the domain whose complement is a convex set. All the results are proved to be sharp.
In this paper, we establish several new versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing $φ=\{φ_n(r)\}^{\infty}_{n=0}$ in place of the $\{r^n\}^{\infty}_{n=0}$ in the power series representations of the functions involved with the Bohr sum and thereby introducing a single parameter, which generalize several related results of earlier authors.
In this paper, we establish five new sharp versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing Schwarz function in place of the initial coefficients in the power series representations of the functions involved and thereby, we generalize several related results of earlier authors.
Bohr type inequalities with one parameter for the class of self analytic maps on unit disc
Ghani Khan Choudhury Institute of Engineering and Technology（Ghani Khan Choudhury工程技术学院）；Gour Mahavidyalaya（高尔学院）；Raiganj University（拉伊甘杰大学）；University of Kalyani（卡利亚尼大学）
In this article, by combining appropriate refined Bohr's inequalities with some techniques concerning bounded analytic functions defined in the unit disk, we generalize and improve several Bohr type inequalities for such functions.
In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.
Recently, there has been a number of good deal of research on the Bohr's phenomenon in various setting including a refined formulation of his classical version of the inequality. Among them, in \cite{PaulPopeSingh-02-10} the authors considered the cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr's inequality for these cases and give a partial answer to a question from \cite{PaulPopeSingh-02-10} for the revised setting.
Let $ \mathcal{H}(\mathbb{D}) $ be the class of analytic functions in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in $ \mathbb{D} $ and $ |f(z)|<1 $ for $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq \frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ 1/3 $ is known as Bohr radius. In this paper, we study Bohr phenomenon for analytic as well as harmonic mappings on simply connected do…
The Bohr radius for the class of harmonic functions of the form $ f(z)=h+\overline{g} $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $, where $ h(z)=\sum_{n=0}^{\infty}a_nz^n $ and $ g(z)=\sum_{n=1}^{\infty}b_nz^n $ is to find the largest radius $ r_f $, $ 0<r_f<1 $ such that \begin{equation*} \sum_{n=1}^{\infty}\left(|a_n|+|b_n|\right)|z|^n\leq d(f(0),\partial f(\mathbb{D})) \end{equation*} holds for $ |z|=r\leq r_f $, where $ d(f(0),\partial f(\mathbb{D})) $ is the Euclidean distance between $ f(0) $ and the boundary of $ f(\mathbb{D}) $. In this paper, we prove two-type of impr…
In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| < 1$, in which we replace the coefficients $a_n$ $(n= 0,1,\ldots ,N)$ of the power series by the derivatives $f^{(n)}(z)/n!$ $(n= 0,1,\ldots ,N)$. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form $f = h + \overline{g}$, where the analytic part $h$ is bounded by $1$ and that $|g'(z)| \le k|h'(z)|$ in $|z| < 1$ and for some $k \in [0,1]$.
In 1914 Bohr proved that there is an $r_0 \in(0,1)$ such that if a power series $\sum_{m=0}^\infty c_m z^m$ is convergent in the open unit disc and $|\sum_{m=0}^\infty c_m z^m|<1$ then, $\sum_{m=0}^\infty |c_m z^m|<1$ for $|z|<r_0$. The largest value of such $r_0$ is called the Bohr radius. In this article, we find Bohr radius for some univalent harmonic mappings having different dilatations and in addition, also compute Bohr radius for the functions convex in one direction.
We determine the Bohr radius for the class of all functions $f$ of the form $f(z)=\sum_{k=1}^\infty a_{kp+m} z^{kp+m}$ analytic in the unit disk $|z|<1$ and satisfy the condition $|f(z)|\le 1$ for all $|z|<1$. In particular, our result also contains a solution to a recent conjecture of Ali, Barnard and Solynin \cite{AliBarSoly} for the Bohr radius for odd analytic functions, solved by the authors in \cite{KayPon1}. We consider a more flexible approach by introducing the $p$-Bohr radius for harmonic functions which in turn contains the classical Bohr radius as special case. Also, we prove sever…
Based on improving the classical Bohr inequality, we get in this paper some refined versions for a quasi-subordination family of functions, one of which is key to build our results. By means of these investigations, for a family of harmonic mappings defined in the unit disk $\D$, we establish an improved Bohr inequality with refined Bohr radius under particular conditions. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. % in a logical way. Here the family of harmonic mappings have the form $f=h+\overline{g}$, where $g(0)=0$, the analytic p…
Let $ \mathcal{H} $ be the class of complex-valued harmonic mappings $ f=h+\bar{g}$ defined in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $, where $ h $ and $ g $ are analytic functions in $ \mathbb{D} $ with the normalization $ h(0)=0=h^{\prime}(0)-1 $ and $ g(0)=0 $. Let $ \mathcal{H}_{0}=\{f=h+\bar{g}\in\mathcal{H} : g^{\prime}(0)=0\}. $ Ghosh and Vasudevrao \cite{Ghosh-Vasudevarao-BAMS-2020} have studied the following interesting harmonic univalent class $ \mathcal{P}^{0}_{\mathcal{H}}(M) $ which is defined by $$\mathcal{P}^{0}_{\mathcal{H}}(M) :=\{f=h+\overline{g} \in \mathc…
Let $ \mathcal{H} $ be the class of harmonic functions $ f=h+\bar{g} $ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C} : |z|<1\}$, where $ h $ and $ g $ are analytic in $ \mathbb{D} $. Let $$\mathcal{P}_{\mathcal{H}}^{0}(α)=\{f=h+\overline{g} \in \mathcal{H} : \real (h^{\prime}(z)-α)>|g^{\prime}(z)|\; \mbox{with}\; 0\leqα<1,\; g^{\prime}(0)=0,\; z \in \mathbb{D}\} $$ be the class of close-to-convex mappings defined by Li and Ponnusamy \cite{Injectivity section}. In this paper, we obtain the sharp Bohr-Rogosinski radius, improved Bohr radius and refined Bohr radius for the class $ \mathcal{P}_{\…
The classical Bohr inequality states that if $ f $ is an analytic function with the power series representation $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disk $ \mathbb{D}:=\{z\in\mathbb{C} : |z|<1\} $ such that $ |f(z)|\leq 1 $ for all $ z\in\mathbb{D} $, then \begin{equation*} \sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\; \text{for}\;\; |z|=r\leq\frac{1}{3} \end{equation*} and the constant $ 1/3 $ cannot be improved. The constant $ r_0=1/3 $ is known as Bohr radius and the inequality $ \sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ is known as Bohr inequality. Let $ \mathcal{H} $ be the class of complex-v…
In this article, we establish the Bohr inequalities for the sense-preserving $K$-quasiconformal harmonic mappings defined in the unit disk $\mathbb{D}$ involving classes of Ma-Minda starlike and convex univalent functions, usually denoted by $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ respectively, and for $\log (f(z)/z)$ where $f$ belongs to the Ma-Minda classes or satisfies certain differential subordination. We also estimate Logarithmic coefficient's bounds for the functions in $\mathcal{C}(ψ)$ for the case $ψ(\mathbb{D})$ be convex.
We prove several improved versions of Bohr's inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $|g'(z)|\le|h'(z)|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. \cite{KayPon2}, for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.
In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic fun…
A class $ \mathcal{F} $ consisting of analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ in the unit disc $ \mathbb{D}=\{z\in\mathbb{C}:|z|<1\} $ satisfies a Bohr phenomenon if there exists an $ r_f>0 $ such that \begin{equation*} I_f(r):=\sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial \mathbb{D}\right) \end{equation*} for every function $ f\in\mathcal{F} $, and $ |z|=r\leq r_f $. The largest radius $ r_f $ is the Bohr radius and the inequality $ I_f(r)\leq{d}\left(f(0),\partial \mathbb{D}\right) $ is Bohr inequality for the class $ \mathcal{F} $, where `$ d $' is the Euclidean distan…
This article focuses on the Bohr radius problem for the derivatives of analytic functions, along with a technique of establishing Bohr inequalities in classical and generalized settings.
We consider the class of all sense-preserving harmonic mappings $f= h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with $g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$ is bounded in $\ID$. $h$ satisfies the condition ${\rm Re}\, h(z)\leq 1$ in $\mathbb{D}$ with $h(0)>0$. both $h$ and $g$ are bounded in $\ID$. $h$ is bounded and $g'(0)=0$. \eee We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of $f$ in $\ID$ is strictly less than $1$. In addition, we determine the…
Bohr-type inequalities for certain integral transforms of operator-valued analytic functions on shifted disc
Ghani Khan Choudhury Institute of Engineering and Technology（Ghani Khan Choudhury 工程与技术学院）；University of North Bengal（北孟加拉大学）；Raiganj University（赖甘杰大学）
We introduce a general class of sense-preserving harmonic mappings defined as follows: \begin{equation*} \mathcal{S}^0_{h+\bar{g}}(M):= \{f=h+\bar{g}: \sum_{m=2}^{\infty}(γ_m|a_m|+δ_m|b_m|)\leq M, \; M>0 \}, \end{equation*} where $h(z)=z+\sum_{m=2}^{\infty}a_mz^m$, $g(z)=\sum_{m=2}^{\infty}b_m z^m$ are analytic functions in $\mathbb{D}:=\{z\in\mathbb{C}: |z|\leq1 \}$ and \begin{equation*} γ_m,\; δ_m \geq α_2:=\min \{γ_2, δ_2\}>0, \end{equation*} for all $m\geq2$. We obtain Growth Theorem, Covering Theorem and derive the Bohr radius for the class $\mathcal{S}^0_{h+\bar{g}}(M)$. As an applicatio…
In this article we prove Bohr inequalities for sense-preserving $K$-quasiconformal harmonic mappings defined in $\mathbb{D}$ and obtain the corresponding results for sense-preserving harmonic mappings by letting $K\to\infty$. One of the results includes the sharpened version of a theorem by Kayumov $\textit{et. al.}$ ($\textit{Math. Nachr.}$, 291 (2018), no. 11--12, 1757--1768). In addition Bohr inequalities have been established for uniformly locally univalent holomorphic functions, and for $\log(f(z)/z)$ where $f$ is univalent or inverse of a univalent function.
Bohr phenomenon for analytic functions $ f $ where $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, first introduced by Harald Bohr in $ 1914 $, deals with finding the largest radius $ r_f $, $ 0<r_f<1 $, such that the inequality $ \sum_{n=0}^{\infty}|a_nz^n|<1 $ holds whenever $ |f(z)|<1 $ holds in the unit disk $ \mathbb{D}=\{z\in\mathbb{C} : |z|<1\} $. The Bohr phenomenon for the harmonic functions of the form $ f(z)=h+\overline{g} $, where $ h(z)=\sum_{n=0}^{\infty}a_nz^n $ and $ g(z)=\sum_{n=1}^{\infty}b_nz^n $ is to find the largest radius $ r_f $, $ 0<r_f<1 $ such that \begin{equation*} \sum_{n=1}^{…
The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0<r_{f}<1$, such that the inequality $\sum_{n=0}^{\infty} |a_{n}z^{n}| \leq 1$ holds whenever the inequality $|f(z)|\leq 1 $ holds in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$. The exact value of this largest radius known as Bohr radius, which has been established to be $r_{f}=1/3$. The Bohr phenomenon \cite{Abu-2010} for harmonic functions $f$ of the form $f(z)=h(z)+\overline {g(z)}$, where $h(z)=\su…
