The Bohr inequality on a simply connected domain and its applications

In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions f in a simply connected domain _, where 0 <1 with a sequence \ₙ (r) \^₍=₀ of non-negative continuous functions defined on [0, 1) such that the series ₍=₀^ₙ (r) converges locally uniformly on [0, 1). Our results represent twofold generalizations corresponding to those obtained for the classes B (D) and B (_), where align* _: =\z C: |z+{1-|<11-\}. align* As a convolution counterpart, we determine the Bohr radius for hypergeometric function on _. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of K -quasiconformal, sense-preserving harmonic maps of the form f=h+g in _.