Harold P. Boas, Dmitry Khavinson
Generalizing a classical one-variable theorem of Harald Bohr, we show that if an n-variable power series has modulus less than 1 in the unit polydisc, then the sum of the moduli of the terms is less than 1 in the polydisc of radius 1/(3*n^{1/2}).
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The Bohr radius for power series of holomorphic functions mapping a multidimensional Reinhardt domain into the convex domain in the complex plane is independent of this convex domain.
The main goal of this paper is to generalize Bohr's phenomenon from complex one-dimensional analysis to higher dimensions in the framework of Quaternionic Analysis.
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。
In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Carathéodory Theorem to the new setting.
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。
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 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.
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.
Based on a generalization of Bohr's equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr's equivalence theorem extended to the case of these functions.
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.
An ordinary Dirichlet series has three abscissae of interest, describing the maximal regions where the Dirichlet series converges, converges uniformly, and con- verges absolutely. The paper of Hille and Bohnenblust in 1931, regarding the region on which a Dirichlet series can converge uniformly but not absolutely, has prompted much investigation into this region, the "Bohr strip". However, a related natural question has apparently gone unanswered: For a Dirichlet series with non-trivial Bohr strip, how far beyond the Bohr strip might the series converge? We investigate this question by explici…
Bohr's theorem for Cesáro operator and certain integral transforms over octonions
本文研究了八元数上片正则函数的塞萨罗算子及伯纳德利、利贝拉和亚历山大算子的玻尔型不等式，证明了相关结果的最优性。
We prove that the abscissas of Bohr and Rogosinski for ordinary Dirichlet series, mapping the right half-plane into the bounded convex domain $G\subset \mathbb{C} $ are independent of the domain $G$. Furthermore, we obtain new estimates about these abscissas.
A bootstrap proof of Bohr's theorem
本文提出Bohr定理的bootstrap新证明，确认最优半径1/3，并给出关于|f(0)|的锐上界，补充Bombieri结果。
A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded complex symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification.
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 note we prove a converse of Bohr's equivalence theorem for Dirichlet series under some natural assumptions.
The Bohr theorem states that any function $f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}$, analytic and bounded in the open unit disk, obeys the inequality $\sum_{n=0}^{\infty} |a_{n}| |z|^{n} < 1$ in the open disk of radius 1/3, the so-called Bohr radius. Moreover, the value 1/$ cannot be improved. In this paper we review some results related to this theorem for the three-dimensional Euclidean space in the setting of quaternionic analysis. The existing results for the Bohr radius will be improved and also some estimates for the hypercomplex derivative of a monogenic function by the norm of the funct…
Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions
该研究考察复 Banach 空间单位球上 $F(x)=f(x)x$ 型全纯映射，建立其几何性质判据，并推广 Bieberbach、Fekete-Szegö、Bloch 与 Alexander 等经典结果。
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 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.
Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of relative Poincaré series associated to loxodromic elements in $Γ$. In complex dimension 2 we describe Bohr-Sommerfeld tori in $Γ\backslash SU(n,1)/U(n)$ associated to hyperbolic elements of $Γ$ and prove that the relative Poincaré series associated to the hyperbolic elements of $…
Bohr Radius and Landau-type Theorems for Harmonic Mappings with Boundary Functions in Lebesgue Spaces
本文研究了由勒贝格空间边界的边界函数诱导的单位盘内调和映射的几何和分析性质，建立了调和映射的亚伯拉罕型不等式，并改进了兰道型定理，确定了单值半径和内切施瓦茨盘半径。
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.
Local Banach Space Theoretic Approach to Bohr's Theorem for Vector Valued Holomorphic and Pluriharmonic Functions
本文通过局部巴拿赫空间理论研究向量值全纯和多值调和函数的波尔定理，证明了波尔半径的正性及渐进行为，并建立了相关系数型施瓦茨-皮克引理。
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.
Given two arbitrary almost periodic functions with associated Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be $^*$-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.
Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications
本文研究K-准共形调和映射在单位圆盘上的广义Bohr不等式，通过引入新的主函数序列改进了传统方法，并推导了新的不等式及Gaussian超几何函数下的Bohr定理扩展。
We prove the following generalisation of Bohr theorem : let $K\subset\mathbb C$ a continuum, $(F_n)_n$ its Faber polynomials, $Ω_R=\{Φ_K<R\}, (R>1)$ the levels sets of the Green function; then there exists $R_0>1$ such that for any $f=\sum_n a_n F_n\in\mathscr O(Ω_{R_0})$ : $f(Ω_{R_0})\subset D(0,1)$ implies $\sum_n|a_n|\cdot|F_n|_K<1$.
An extension of Bohr's theorem
本文推广了Bohr定理，证明若Dirichlet级数可解析延拓到半平面并避开两个不同复数值，则其在任意更小半平面上一致收敛，且该条件是最优的。
A note on Bohr's theorem for Beurling integer systems
本文研究Beurling整数系统中Bohr定理的成立条件，证明对Beurling素数序列施加小扰动可使其满足Bohr条件，并结合概率方法构造出Bohr定理与Riemann假设同时成立的系统，从而给出Helson关于广义Dirichlet级数Hardy空间外函数猜想的反例。
Coefficient estimates and Bohr phenomenon for pluriharmonic mappings in the polydisc
该研究在多复变量框架下引入正则化多重调和映射类，建立其精确系数估计与增长定理，确定Bohr半径并研究函数截面，推广了已有结果。
We use an observation of Bohr connecting Dirichlet series in the right half plane $\mathbb{C}_+$ to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra $\mathcal…
On the Phragmen Lindelöf Theorem in strips
本文研究了多项式有界解析函数在条带中的L²k-类Bohr可和性泛函。证明了k中的近似凹性等价于阶函数的递增性。对于扩展Selberg类中的L函数，这表明阶函数处处可微且在一边有二次衰减。还建立了严格凹性与Lindelöf假设之间的联系。
For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
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