Bohr-type inequalities for certain integral transforms of operator-valued analytic functions on shifted disc

In this paper, we investigate Bohr-type inequalities for certain integral transforms of some bounded holomorphic operator-valued functions defined on a shifted disc. More precisely, we consider functions $f\in H^{\mathrm{\infty }}({\mathrm{\Omega }}_{\gamma },\mathcal{B}(\mathcal{H}))$, where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded linear operators on a complex Hilbert space $\mathcal{H}$, and study the Bohr phenomena associated with the series representations in \eqref{e1.2a} and \eqref{e2.1}. Moreover, we determine the sharp radii $R_{\gamma }>0$ for which the corresponding majorant series satisfy the required inequalities for all $\mathrm{\mid }z\mathrm{\mid }\le R_{\gamma }$. One of our main results extends some recent result due to Kumar and Sahoo \cite{Kumar-Sahoo-Meditarr-2023} to a large extent, and the other one is new in the literature