Bohr operator on operator-valued polyanalytic functions on simply connected domains

Abstract

AbstractIn this article, we study the Bohr operator for the operator-valued subordination class$S(f)$consisting of holomorphic functions subordinate tofin the unit disk$\mathbb {D}:=\{z \in \mathbb {C}: |z|<1\}$, where$f:\mathbb {D} \rightarrow \mathcal {B}(\mathcal {H})$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 analogs of a couple of interesting subordination results from scalar-valued settings. We also obtain a von Neumann-type inequality for the class of analytic self-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}$.