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

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 $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$, in a simply connected domain \begin{align*} Ω_γ=\biggl{z\in$\mathbb{C}$: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ};\mbox{for}; 0\leq γ<1\biggr}, \end{align*} where ${\mathrm{\Omega }}_0=\mathbb{D}$. 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 $f$.