Bohr radius for invariant families of bounded analytic functions and certain Integral transforms

Abstract: In this paper, we first obtain a refined Bohr radius for invariant families of bounded analytic functions on unit disk $ \mathbb{D} $. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, in a simply connected domain \begin{align*} \Omega_\gamma=\biggl{z\in\mathbb{C}: \bigg|z+\dfrac{\gamma}{1-\gamma}\bigg|<\dfrac{1}{1-\gamma}\;\mbox{for}\; 0\leq \gamma<1\biggr}, \end{align*} where $ \Omega_0=\mathbb{D} $. These results generalize some existing results. We also show that a better estimate can be obtained in radius and inequality can be shown sharp for Laplace transform of $ f $.