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In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.

Complex Variables · Mathematics 2025-02-06 Jianying Zhou , Wanqing Hou , Boyong Long

We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality.…

Functional Analysis · Mathematics 2011-07-08 Masatoshi Fujii , Mohammad Sal Moslehian , Jadranka Micic

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…

Complex Variables · Mathematics 2019-04-01 Seraj A. Alkhaleefah , Ilgiz R Kayumov , Saminathan Ponnusamy

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on…

Complex Variables · Mathematics 2026-03-18 Vasudevarao Allu , Raju Biswas , Rajib Mandal

Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new…

Complex Variables · Mathematics 2024-09-17 Shanshan Jia , Ming-Sheng Liu , Saminathan Ponnusamy

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{D}^n$.We also prove two other sharp versions of the Bohr inequality in the setting…

Complex Variables · Mathematics 2025-12-19 Molla Basir Ahamed , Sujoy Majumder , Nabadwip Sarkar

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{P}\Delta(0;1_n)$. We also prove two other sharp versions of the Bohr inequality in…

Complex Variables · Mathematics 2025-12-09 Molla Basir Ahamed , Sujoy Majumder , Nabadwip Sarkar , Ming-Sheng Liu

In this paper, we first obtain a refined Bohr radius for invariant families of bounded analytic functions on unit disk $ \mathbb{D} $. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace…

Complex Variables · Mathematics 2024-05-08 Molla Basir Ahamed , Partha Pratim Roy , Sabir Ahammed

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…

Algebraic Geometry · Mathematics 2007-05-23 Rouchdi Bahloul

The main aim of this paper is to establish several Landau-type theorems for certain bounded poly-analytic functions and reduced poly-analytic functions that generalize some previously established results.

Complex Variables · Mathematics 2025-08-28 Vasudevarao Allu , Raju Biswas , Rajib Mandal , Hiroshi Yanagihara

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Ces\'aro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to…

Complex Variables · Mathematics 2026-04-14 Vasudevarao Allu , Raju Biswas , Rajib Mandal

In this paper, we present some double inequalities involving certain ratios of the Gamma function. These results are further generalizations of several previous results. The approach is based on the monotonicity properties of some functions…

Classical Analysis and ODEs · Mathematics 2015-06-25 Kwara Nantomah

In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.

Algebraic Topology · Mathematics 2018-07-24 Ryuma Orita

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $ f\in\mathcal{B} $, then the…

Complex Variables · Mathematics 2023-08-28 Molla Basir Ahamed

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect…

Classical Analysis and ODEs · Mathematics 2024-11-21 Dmitrii Karp , Elena Prilepkina

This paper introduces a unified framework for Bohr-type inequalities by incorporating multiple Schwarz functions into the majorant series for $K$-quasiconformal harmonic mappings in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| <…

Complex Variables · Mathematics 2025-10-02 Raju Biswas , Rajib Mandal

This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier…

Functional Analysis · Mathematics 2020-10-07 Eskil Rydhe

In this article, we first establish operator-valued analogues of multidimensional refined Bohr inequality. Then we establish operator-valued analogues of multidimensional improved Bohr inequality with a certain power of the norm of the…

Complex Variables · Mathematics 2023-08-29 Sabir Ahammed , Molla Basir Ahamed

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…

Complex Variables · Mathematics 2018-01-11 J. M. Sepulcre , T. Vidal

In this paper, we study Bohr's inequality and refined versions of Bohr-Rogosinski inequalities involving Schwarz functions. Moreover, we establish a version of multidimensional analogue of Bohr inequality and Bohr-Rogosinski inequalities…

Complex Variables · Mathematics 2023-12-12 Sabir Ahammed , Molla Basir Ahamed