Amir Ismagilov, Ilgiz R Kayumov, Saminathan Ponnusamy
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.
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We prove that the Bohr' radius for large functions is $e^{-π}.$
Bohr inequality and Bohr-Rogosinski inequality for $K$-Quasiconformal harmonic mappings
本文研究单位圆盘上解析部分从属于凹单叶函数类、Ma-Minda凸与星形函数类的$K$-拟共形保向调和映射，证明了若干精确的Bohr型及Bohr-Rogosinski型不等式，推广了已有结果。
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.
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.
The Bohr inequality for vector-valued holomorphic functions with lacunary series in complex Banach spaces
本文研究有限维Banach序列空间单位球内带缺项级数的向量值全纯函数的Bohr不等式与Bohr-Rogosinski不等式，且证明所有结果均为精确的。
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.
Refined Bohr inequalities and a refined Bohr-Rogosinski inequality on complex Banach spaces
本文针对复Banach空间单位球上的全纯函数类，建立了精细版Bohr不等式、泛函型与范数型精细Bohr不等式、Bohr-Rogosinski不等式及改进版Bohr不等式，并证明所有结果均精确。
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.
Sharpness in Bohr's Inequality
本文沿 Kayumov 和 Ponnusamy 的路线分析玻尔不等式，通过在右端添加依赖函数的额外加项，研究较小半径下的锐常数，并改进了若干已有结果。
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 $.…
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…
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.
Multidimensional analogues of the improved Bohr's inequality
本文研究了多维情况下的玻尔不等式改进，通过替换常数项为函数绝对值和绝对值平方，提出了两个严格版本。
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.
Bohr phenomenon for analytic and harmonic mappings on shifted disks
本文研究了移位圆盘上分析函数和调和映射的Bohr不等式及其改进形式，推导了多个精确结果。
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…
Operator-valued analogues of multidimensional Bohr-Rogosinski inequalities
该研究针对多维Bohr不等式相关问题，建立了四类不同形式的多维Bohr不等式的算子值类似物，且证明了所有结果均为精确的。
Bohr type inequality for certain integral operators and Fourier transform on shifted disks
本文推导了Césàro算子、Bernardi积分算子和离散傅里叶变换在移位圆盘上定义的有界解析函数上的精确Bohr型不等式。
Multidimensional analogues of the improved Bohr's inequality for shifted polydisks
研究\(\mathbb{C}^n\)中一般单连通域上全纯函数的玻尔现象，改进相关现有结果，还考虑含单位多圆盘的多圆盘上多重调和映射，建立玻尔型不等式。
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…
The Bohr radius for operator valued functions on simply connected domain
本文在单连通域上改进并推广了算子值全纯函数及ν-Bloch函数的Bohr不等式与Bohr-Rogosinski不等式，所得结果均为最优。
The Bohr-type inequalities for holomorphic functions with lacunary series in complex Banach space
本文研究了有限维Banach序列空间单位球中单值和向量值全纯函数的缺项级数Bohr不等式，并将交错级数的Bohr不等式扩展至高维空间。
Sharp Bohr-Type inequalities for certain classes of close-to-convex functions
本文确定了开单位圆盘上某些近凸函数子类的Rogosinski半径，并建立了这些子类的经典Bohr不等式和Bohr-Rogosinski不等式的改进版本，所有结果都是尖锐的。
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…
Bohr phenomena for slice regular functions over Quaternions
本文针对四元数上的切片正则函数，建立了切片星形、近凸函数的玻尔不等式，给出其推广与改进版本，并提供满足实部条件的切片正则函数玻尔不等式精细化形式，且所有结果最优。
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…
Bohr-Rogosinski inequalities involving Schwarz functions
本文研究涉及 Schwarz 函数的 Bohr 及 Bohr-Rogosinski 不等式，给出其加细形式与多维推广，并证明所有结果均为精确。
Sharp Bohr and Bohr-Rogosinski inequalities involving area measure for close-to-convex harmonic mappings
本文针对近凸调和映射子类，通过引入与面积积分相关的单调函数，建立了用欧几里得距离表示的尖锐Bohr型不等式及含模幂的Bohr-Rogosinski型不等式，并证明半径尖锐性及识别极值函数，推广统一了多个经典结果。
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.
Bohr inequalities for holomorphic mappings in higher-dimensional complex Banach spaces
本文在高维复Banach空间中研究全纯映射的Bohr现象，推导了改进的Bohr不等式，并证明了所得到的Bohr半径的精确性。
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.
Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk
本文针对单位圆盘上的自解析映射类与调和映射类，通过引入子圆盘面积测度、高阶导数量及连续函数序列，建立了经典 Bohr 不等式的多种改进与精细化版本，并给出应用。
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.
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 $' is the Euclidean distan…
The Bohr's Phenomenon for the class of K-quasiconformal harmonic mappings
本文研究K-准共形保形调和映射类中的改进玻尔不等式及其精炼版本，通过引入非负量S_ρ(h)和替代主项系数的方法，获得单位圆盘内调和映射的精确玻尔-罗戈辛斯基半径。
Improved and Refined Bohr-Type Inequalities for Slice Regular Functions over Octonions
本文研究八元数上切片正则函数的 Bohr 型不等式，给出其推广、改进与精细化版本，并证明所有结果都是精确的。
A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk
该研究统一探讨单位圆盘上解析与调和映射的Bohr不等式改进形式，引入可微递增函数序列及Hurwitz-Lerch Zeta函数，所得结果均为最优。
Bohr-Rogosinski Inequalities for bounded analytic functions
本文改进了针对有界解析函数类$\mathcal{B}$的玻尔-罗戈辛斯基不等式，引入了子圆面积测度以提升不等式的精度。
Bohr-type inequalities for classes of analytic maps and K-quasiconformal harmonic mappings
本文通过结合经典Bohr不等式的改进版本与面积测度方法，推广并精化了解析自映射类及K-拟共形调和映射类的Bohr型不等式，并证明结果最优。
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.
Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n
本文研究了多复变量中Schwarz函数和Euler算子的锐Bohr-Rogosinski半径，通过多维Bohr不等式推广和导数估计，确定了最优半径并验证了其锐性。
The Bohr inequality on a simply connected domain and its applications
本文在单连通区域上建立广义玻尔不等式并证明其尖锐性，推广了单位圆盘及相关区域上的经典结果，同时确定了超几何函数的玻尔半径，并进一步将该不等式推广到K-拟共形保向调和映射类。
Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator
本文研究了解析函数类A_β的Bohr现象及尖锐系数问题，通过整合复动力学和几何函数理论，推导出经典Bohr半径的尖锐改进，并建立了广义Bohr和Bohr-Rogosinski不等式，同时为Fekete-Szegö问题提供了尖锐解。
The Bohr radius for a class $\mathcal{G}$ consisting of analytic functions $f(z)=\sum_{n=0}^{\infty}a_nz^n$ in unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ is the largest $r^*$ such that every function $f$ in the class $\mathcal{G}$ satisfies the inequality \begin{equation*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right) = \sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})) \end{equation*} for all $|z|=r \leq r^*$, where $d$ is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions $f$ satisfying differential subor…
Sharp Bohr-Type Inequalities Involving Euler Operator and Area Functionals on $\mathbb{P}Δ(0;1_n)$
该研究针对多复变量的单位多圆盘上的有界全纯函数，建立并改进了含欧拉算子、面积泛函的尖锐玻尔不等式，将相关结果推广至多维情形并证明其尖锐性。
This paper examines the coefficient problems for the class of semigroup generators, a topic in complex dynamics that has recently been studied in context of geometric function theory. Further, sharp bounds of coefficient functional such as second order Hankel determinant, third order Toeplitz and Hermitian-Toeplitz determinants are derived. Additionally, the sharp growth estimates and the bounds of difference of successive coefficients are determined, which are used to prove the Bohr and the Bohr-Rogosinski phenomenon for the class of semigroup generators.
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…
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.
The sharp refined Bohr inequalities for a subclass of close-to-convex harmonic mappings
本文研究了一类特定调和映射子类的波尔现象、细化波尔半径及波尔-罗戈辛斯基不等式，通过引入新的条件和参数，得出了更精确的不等式结果。
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