Bohr Inequality Research Articles

Some Refinements and Generalizations of Bohr’s Inequality

In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, as well as leveraging the class of functions defined by the Daykin–Eliezer–Carlitz inequality. Our novel contribution lies in demonstrating that Bohr’s and Bergström’s inequalities can be derived from one another, revealing a deeper interconnectedness between these results. Furthermore, we present several new generalizations of Bohr’s inequality, along with other notable inequalities from the literature, and discuss their various implications. By providing more comprehensive and verifiable conditions, our work extends previous research and enhances the understanding and applicability of Bohr’s inequality in mathematical studies.

Operator Bohr-type inequalities

Abstract The classical Bohr inequality for scalars was extended to the non-commutative case of Hilbert space operators in the literature. The sole goal of this article is to discuss the operator Bohr inequality and present some of its new variants. This includes fresh reverses and refinements of this inequality with applications towards an operator’s real and imaginary parts, not to forget the new discussion of different domains of the parameters. One further application towards the operator Dunkl-Williams inequality will be presented too. While the new results are interesting, we emphasize that the approach used to explore these inequalities differs from the existing literature methods for this context.

Bohr Inequalities for Certain Classes of Harmonic Mappings

Bohr Inequalities for Certain Classes of Harmonic Mappings

On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Abstract The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

Harmonic analogue of Bohr phenomenon of certain classes of univalent and analytic functions

A class $ \mathcal {F} $ consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C}:\lvert z\rvert <1\} $ is said to satisfy Bohr phenomenon if there exists an $ r_f>0 $ such that $$ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial \mathbb {D}) $$ for every function $ f\in \mathcal {F} $, and $\lvert z\rvert =r\leq r_f $. The largest radius $ r_f $ is known as the Bohr radius and the inequality $ \sum _{n=1}^{\infty }\lvert a_n\rvert r^n\leq d(f(0),\partial f(\mathbb {D})) $ is known as the Bohr inequality for the class $ \mathcal {F} $, where $d$ is the Euclidean distance. In this paper, we prove several sharp improved and refined versions of Bohr-type inequalities in terms of area measure of functions in a certain subclass of analytic and univalent (i.e. one-to-one) functions. As a consequence, we obtain several interesting corollaries on the Bohr-type inequality for the class which are the harmonic analogue of some Bohr-type inequality for the class of analytic functions.

Refined Bohr inequality for functions in and in complex Banach spaces

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 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 in the unit disk . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions defined on a balanced domain . All the results are proved to be sharp.

Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex

AbstractIn this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk $ \mathbb {D} $ , class of analytic functions $ f $ defined on $ \mathbb {D} $ such that $\mathrm {Re}\left (f(z)\right )<1 $ , and class of subordination to a function g in $ \mathbb {D} $ . Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

AbstractLet be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ2. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in and are sharp in the case with values in ℓ2.

Stable classes of harmonic mappings

Let H0 denote the set of all sense-preserving harmonic mappings f=h+g‾ in the unit disk D, normalized with h(0)=g(0)=g′(0)=0 and h′(0)=1. In this paper, we mainly investigate some properties of certain subclasses of H0, including inclusion relations and stability analysis by precise examples, coefficient bounds, growth, covering and distortion theorems. As applications, we build some Bohr inequalities for these subclasses by means of subordination. Among these subclasses, six classes consist of functions f=h+g‾∈H0 such that h+ϵg is univalent (or convex) in D for each |ϵ|=1 (or for some |ϵ|=1, or for some |ϵ|≤1). Simple analysis shows that if the function f=h+g‾ belongs to a given class from these six classes, then the functions h+ϵg‾ belong to corresponding class for all |ϵ|=1. We call these classes as stable classes.

Bohr's phenomenon for holomorphic and harmonic functions with lacunary series in complex Banach spaces

We determine the sharp Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and takes values from the unit disk. Also, we investigate the Bohr-type inequalities for harmonic functions f = h + g ¯ defined on a balanced domain G ⊂ X .

The sharp refined Bohr–Rogosinski inequalities for certain classes of harmonic mappings

A class F consisting of analytic functions f ( z ) = ∑ n = 0 ∞ a n z n in the unit disc D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an 0 $ ]]> r f > 0 such that ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) for every function f ∈ F , and | z | = r ≤ r f . The largest radius r f is the Bohr radius and the inequality ∑ n = 1 ∞ | a n | r n ≤ d ( f ( 0 ) , ∂f ( D ) ) is Bohr inequality for the class F , where ‘d’ is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings.

Improved Bohr inequality for harmonic mappings

AbstractIn order to improve the classical Bohr inequality, we explain some refined versions for a quasi‐subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk . Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form , where , the analytic part h is bounded by 1 and that in and for some .

Refinement of Powered Bohr Inequality

Refinement of Powered Bohr Inequality

A generalization of the Bohr inequality and its applications

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 inequality for the class of bounded analytic functions defined on the simply connected domain Part of its applications, we calculate the Bohr-type radii for some known integral operators.

On the Bohr and Selberg inequalities in 2-*-inner product spaces

The Bohr inequality occurs for scalars, vectors, matrices and operators. In this paper, we introduce the new type of Bohr's inequality in Hilbert -modules and in 2--inner product spaces, which is equipped by a 2-inner product map that takes values on locally -algebras. Then, we obtain a version of the Selberg and Bessel inequality and its results in an -2-inner product space.

The Bohr Inequality for the Generalized Cesáro Averaging Operators

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.

The Bohr inequality for certain harmonic mappings

Let HC(ϕ) and HCc(ϕ) denote the classes of sense-preserving harmonic mappings f=h+g¯ in D with the dilation g′(z)=αzh′(z) for |α|<1 such that h is Ma–Minda type convex function and Ma–Minda type convex function with respect to the conjugate points respectively. Here, the function ϕ:D→ℂ, called Ma–Minda function, is analytic and univalent in D such that ϕ(D) has positive real part, symmetric with respect to the real axis, starlike with respect to ϕ(0)=1 and ϕ′(0)>0. The classes HC(ϕ) and HCc(ϕ) are derived from the work of Sun et al. (2016) on the class M(α,β). We study the growth theorems for the functions in these classes. For general ϕ, the sharp coefficient bounds of the functions in the classes HC(ϕ) and HCc(ϕ) are not yet known. For the function f=h+g¯ of the form h(z)=z+∑n=2∞anzn and g(z)=∑n=2∞bnzn in D, using the Bohr phenomenon for subordination classes, we find the radius Rf<1 such that Bohr inequality |z|+∑n=2∞(|an|+|bn|)|z|n≤d(f(0),∂f(D))holds for |z|=r≤Rf for functions in the classes HC(ϕ) and HCc(ϕ) . As a consequence, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes. In addition, we discuss the area theorem for functions in the class HC(ϕ) and obtain an improved version of Bohr inequality for functions in HC(ϕ).

Refinements of the Bohr and Rogosinski phenomena

We obtain improved versions of a classical theorem of Rogosinski concerning the partial sums of a bounded holomorphic function defined on the open unit disk D. Further, we establish refined versions of a generalized Bohr inequality for holomorphic functions mapping D inside a simply connected or convex domain Ω⊊C. In addition, we improve on the classical Bohr inequality for the family of holomorphic self mappings of D and for its subfamily consisting of functions that fix the origin.

Improved Bohr inequalities for certain class of harmonic univalent functions

Let be the class of complex-valued harmonic mappings defined in the unit disk , where h and g are analytic functions in with the normalization and . Let Let be the class of harmonic univalent mappings in the unit disk [Ghosh N, Allu V. On some subclasses of harmonic mappings. Bull Aust Math Soc. 2020;101:130–140.]. In this paper, we obtain the sharp Bohr–Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class .

Bohr’s inequality for non-commutative Hardy spaces

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n, n\ge 2$.