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 Ω_γ, where 0≤ γ<1 with a sequence \φₙ(r) \∞ₙ₌₀ of non-negative continuous functions defined on $[0,1)$ such that the series ∑ₙ₌₀∞φₙ(r) converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes B(D) and B(Ωγ), where {align*} Ωγ:=\{z∈ C: |z+γ/1-γ|<1/1-γ\}. {align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on Ωγ. 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+ḡ in Ωγ.