Bohr operator on opertor valued polyanalytic functions on simply connected domains

Abstract

In this article, we study the Bohr operator for the operator valued subordination class $S(f)$ consisting of holomorphic functions subordinate to $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, where $f:\mathbb{D}\to \mathcal{B}(\mathcal{H})$ is holomorphic and $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal{H}$. We establish several subordination results, which can be viewed as the analogues of a couple of interesting subordination results from scalar valued settings. We also obtain a von Neumann-type inequality for the class of self-analytic mappings of the unit disk $\mathbb{D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator valued polyanalytic functions in certain proper simply connected domains in $\mathbb{C}$. We obtain Bohr radius for the operator valued polyanalytic functions of the form $F(z)={\sum }_{l=0}^{p-1}{\overline{z}}^l{f}_l(z)$, where $f_0$ is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk $\mathbb{D}$.

We revise the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in this article. 11 pages

References in corpus (1)

• The Bohr Phenomenon for analytic functions on simply connected domains