  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
$f:\mathbb{D} \rightarrow \mathcal{B}(\mathcal{H})$ 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 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 $\mathbb{D}$.
