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 part $h$ is bounded by 1 and that $|g'(z)|\leq k|h'(z)|$ in $\D$ and for some $k\in[0,1]$.
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.
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, 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.
This article is devoted to the sharp improvement of the classical Bohr inequality for bounded analytic functions defined on the unit disk. We also prove two other sharp versions of the Bohr inequality by replacing the constant term by the absolute of the function and the square of the absolute of the function, respectively.
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 article, we study the Bohr type inequality for {C}esáro operator and {B}ernardi integral operator acting on the space of analytic functions defined on a simply connected domain containing the unit disk $\mathbb{D}$.
The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator.
In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing $\mathbb{D}$ and obtain a Bohr type inequality.
In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions.
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…
Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. In this article, we study a generalized Bohr's inequality for the class of bounded analytic functions defined on the simply connected domain $$ Ω_γ:=\bigg\{z\in \mathbb{C}: \bigg|z+\fracγ{1-γ}\bigg|<\frac{1}{1-γ}\bigg\}, \,\ \text{for } 0\leq γ<1. $$ Part of its applications, we calculate the Bohr-type radii for some known integral operators.
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 $.…
We determine the Bohr radius for the class of odd functions $f$ satisfying $|f(z)|\le 1$ for all $|z|<1$, settling the recent conjecture of Ali, Barnard and Solynin \cite{AliBarSoly}. In fact, we solve this problem in a more general setting. Then we discuss Bohr's radius for the class of analytic functions $g$, when $g$ is subordinate to a member of the class of odd univalent functions.
In this paper, first we give a new generalization of the Bohr's inequality for the class of bounded analytic functions $\mathcal{B'}$ and for the class of sense-preserving $K$-quasiconformal harmonic mappings of the form $f=h+\overline{g},$ where $h \in \mathcal{B'}.$ Finally we give a new generalization of the Bohr's inequality for the class of analytic functions subordinate to univalent functions and for the class of sense-preserving $K$-quasiconformal harmonic mappings of the form $f=h+\overline{g},$ where $h$ is subordinated to some analytic function.
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.
The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. Thes…
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 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.
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…
This paper mainly uses the nonnegative continuous function $\{ζ_n(r)\}_{n=0}^{\infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $\real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and even bits of $f(z)$ are also established. We will prov…
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…
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}_{\…
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.
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 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…
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…
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…
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…
The object of this paper is to study the powered Bohr radius $ρ_p$, $p \in (1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=\sum_{k=0}^\infty |a_k|^p r^k$, then we show that $M_p^f (r)\leq 1$ for $r \leq r_p$ where $r_ρ$ is the powered Bohr radius for conformal automorphisms of the unit disk. This answers the open problem posed by Djakov and Ramanujan in 2000. A couple of other consequences of our approach is also stated, including an asymptotically sharp form of one of the results of Djakov…
The classical inequality of Bohr concerning Taylor coeficients of bounded holomorphic functions on the unit disk, has proved to be of significance in answering in the negative the conjecture that if the non-unital von Neumann inequality held for a Banach algebra then it was necessarily an operator algebra. Here we provide a rather short and easy proof of the inequality.
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…
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.
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 $k \in [0,1]$.
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.
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}^{…
