In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.
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.
Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces
Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）；Indian Institute of Science（印度科学研究所）；National Institute of Science Education and Research（国家科学教育研究中心）
Generalizing the classical result of Bohr, we show that if an n-variable power series converges in an n-circular bounded complete domain D and its sum has modulus less than 1, then the sum of the maximum of the moduli of the terms is less than 1 in the homothetic domain r*D, where r = 1 - (2/3)^(1/n). This constant is near to the best one for the domain D = {z: |z_1| + ... + |z_n| < 1}.
In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $\mathbb{C}^n$ under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain $ G $ of a complex Banach space $ X $ and take values from the unit disk $ \mathbb{D} $. Furthermore, as a consequence of one of this results, we obtain a refined version of the Bohr-type inequalities for harmonic functions $ f=h+\bar{g} $ defined on a balanced domain $ G\subset X $.…
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.
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…
Recently the present authors established refined versions of Bohr's inequality in the case of bounded analytic functions. In this article, we state and prove a generalization of these results in a reformulated "distance form" version and thereby we extend the refined versions of the Bohr inequality for the class of the quasi-subordinations which contains both the classes of majorization and subordination as special cases. As a consequence, we obtain several new results.
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…
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.
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.
In this paper, firstly we prove two refined Bohr-type inequalities associated with area for bounded analytic functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the unit disk. Later, we establish the Bohr-type operator on analytic functions and sections.
The concept of Bohr radius for the class of bounded analytic functions was introduced by Harald Bohr in 1914. His initial result received greater interest and was sharpened-refined-generalized by several mathematicians in various settings--which is now called Bohr phenomenon. Various generalization of Bohr's classical theorem is now an active area of research and has been a source of investigation in numerous other function spaces and including holomorphic functions of several complex variables. Recently, a new generalization of Bohr's ideas was introduced and investigated by Kayumov et al.. I…
The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to $1$, then the sum of absolute values of its terms is less than or equal to $1$ for the subdisk $|z|<1/3$ and $1/3$ is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present a refined version of Bohr's inequality along with few other related improved versions of previously known results.
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…
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…
We consider the class of \emph{stable} harmonic mappings $f=h+\overline{g}$ introduced by Martin, Hernandez, and the class of \emph{stable} logharmonic mappings $f=zh\overline{g}$ introduced by AbdulHadi, El-Hajj. We determine Bohr's radius for the classes of stable univalent harmonic mappings, stable convex harmonic mappings and stable univalent logharmonic mappings. We also consider improved and refined versions of Bohr's inequality and discuss the Bohr's Rogonsiski radius for these family of mappings.
