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The Bohr radius for operator valued functions on simply connected domain

Sabir Ahammed, Molla Basir Ahamed

Abstract

In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions $f$ on a simply connected domain $Ω$ in $\mathbb{C}$. Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence $\varphi=\{\varphi_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions in $[0,1)$ such that the series $\sum_{n=0}^{\infty}\varphi_n(r)$ converges locally uniformly on the interval $[0,1)$. All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued $ν$-Bloch functions defined in two different simply connected domains, $Ω$ and $Ω_γ$, in $\mathbb{C}$.

The Bohr radius for operator valued functions on simply connected domain

Abstract

In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions on a simply connected domain in . Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence of non-negative continuous functions in such that the series converges locally uniformly on the interval . All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued -Bloch functions defined in two different simply connected domains, and , in .

Paper Structure

This paper contains 9 sections, 16 theorems, 177 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The Bohr inequality on a simply connected domain
  3. The Bohr inequality involving a sequence $\varphi=\{\varphi_n(r) \}^{\infty}_{n=0}$ of non-negative continuous function in $[0,1)$
  4. Proof of Theorems \ref{['th-3.3']}, \ref{['th-3.4']}, \ref{['th-3.5']}, \ref{['BS-thm-4.9']}, and \ref{['BS-thm-5.1']}
  5. Generalization of the Bohr inequality for the class $\mathcal{B}\left(\Omega_{\gamma}\right)$
  6. Application of Theorem \ref{['BS-thm-3.1']}
  7. Bohr inequality for Bloch spaces on simply connected domain
  8. Concluding remarks
  9. Declarations

Key Result

Theorem 2.1

Let $\Omega$ be a simply connected domain containing the unit disk $\mathbb{D}$ and $f\in {H}^{\infty}\left(\Omega, \mathcal{B}(\mathcal{H})\right)$ be given by $f(z)=\sum_{n=0}^{\infty}A_nz^n$ in $\mathbb{D}$. Then where $S_r$ denotes the area of the image of the disk $\mathbb{D}(0,r)$ under the mapping $f$ and $P(w)$ is a polynomial given by e-2.3a. The equality $\mathcal{A}_{\lambda_H}^f(r)=1$

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.1
  • Corollary 2.2
  • ...and 20 more