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Operator valued analogues of multidimensional refined Bohr's inequalities

Vasudevarao Allu, Raju Biswas, Rajib Mandal

Abstract

Let $\mathcal{B}(\mathcal{H})$ denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces $\mathcal{H}$. In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class $\mathcal{B}(\mathbb{D}, \mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}$ to $\mathcal{B}(\mathcal{H})$ with $\sup_{|z|<1}\Vert f(z)\leq 1$ by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain $Ω\subset\mathbb{C}^n$. All of the results are sharp.

Operator valued analogues of multidimensional refined Bohr's inequalities

Abstract

Let denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces . In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class of bounded analytic functions from the unit disk to with by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain . All of the results are sharp.

Paper Structure

This paper contains 7 sections, 16 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Bohr theorem for operator valued analytic function
  3. Operator valued analogues of multidimensional Bohr inequality
  4. key lemmas
  5. main Results
  6. Proofs of the main results
  7. Statements & Declarations

Key Result

Lemma 4.1

10 Let $B(z)$ be an analytic function with values in $\mathcal{B}(\mathcal{H})$ and satisfying $\Vert B(z)\Vert \leq 1$ on $\mathbb{D}$. Then for each $a\in\mathbb{D}$ and $n\in\mathbb{N}$.

Figures (3)

  • Figure 1: The graph of the polynomial $G_5(r)$ for $r\in\left(0,1\right)$
  • Figure 2: The graph of $G_6(a_1,r)$ for $r>r_1$
  • Figure 3: The graph of the polynomials $\Phi(0,r)$ and $\Phi(1,r)$ for $0\leq r<1$

Theorems & Definitions (32)

  • Definition 3.1
  • Definition 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Remark 4.1
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 22 more