In this paper, we obtain Bohr-type inequalities for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded holomorphic functions \( f\in H^{\infty }\left( \Omega _{\gamma }, \mathcal {B} (\mathcal {H})\right) \) be given by \( f(z)=\sum _{n=0}^{\infty }A_nz^n \) in \( \mathbb {D} \) with \(||f(z)||_{H^{\infty }\left( \Omega _{\gamma }, \mathcal {B} (\mathcal {H})\right) }\le 1,\) where \(A_n\in \mathcal {B} (\mathcal {H})\) for \(n\in \mathbb {N}_0:=\mathbb {N}\cup \{0\}\) and \(||A_0||<1,\) and \(\begin{aligned} \Omega _\gamma :=\biggl \{z\in \mathbb {C}: \bigg |z+\dfrac{\gamma }{1-\gamma }\bigg |<\dfrac{1}{1-\gamma }\;\text{ for }\; 0\le \gamma <1\biggr \}. \end{aligned}\) Moreover, we establish the Bohr–Rogosinski inequality for Cesáro operator on the class \( H^{\infty }\left( \mathbb {D}, \mathcal {B} (\mathcal {H})\right) \) of holomorphic functions f. All the results are proved to be sharp.
