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 D. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded analytic functions f (z) =₍=₀^aₙzⁿ, in a simply connected domain align* _=\z: |z+{1-|<11-\;for\; 0 <1\}, align* where ₀=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.