On the Bohr Inequalities for Certain Integral Transforms

We consider analytic functions of the form \(f(z)=\sum _n{a_n z^n}\) with \(|f(z)|\le 1\) defined on the unit disc \(\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}\). Due to studies on the Bohr phenomenon concerning this class of functions, and recent results on the Bohr inequality of some integral operators, we are interested in Bohr inequalities pertaining to integral transforms. We first obtain a Bohr-type inequality for the (discrete) Fourier transform acting on the functions f defined above, alongside the associated Bohr radius. We find that this inequality is sharp, and that the constant dictating the Bohr radius cannot be improved. We obtain a secondary result by finding an expression for \(a:=|a_0|\) that maintains the Bohr inequality even if \(r:=|z|\) grows past the Bohr radius. We also investigate the behaviour of the Fourier transform of f as \(r\rightarrow 1\), by finding the limiting bound for the aforementioned transform. We prove that this bound is actually also sharp. We then study the (discrete) Laplace transform of f and obtain its relevant Bohr inequality.