Bohr Radius and Landau-type Theorems for Harmonic Mappings with Boundary Functions in Lebesgue Spaces

Abstract

This paper investigates the geometric and analytical properties of harmonic mappings $f$ in the unit disk $\mathbb{D}$ induced by boundary functions $F$ belonging to the Lebesgue spaces $L^{p}(\mathbb{T})$ for $1 \le p \le \infty$. We first establish a sharp Bohr-type inequality for the class of bounded harmonic mappings. Specifically, we prove that for a fixed analytic part $|a_{0}|= aM$, the majorant series $M_{f}(r)$ satisfies $M_{f}(r) \le M$ for $r \le (1-a)/(1-a+4/\pi)$, and demonstrate that this radius is best possible. This result is subsequently extended to harmonic mappings with $L^p$ boundary functions, where we determine the sharp Bohr radius $r_{p} = 1/(2C_{q}+1)$, with $C_{q}$ being a constant depending on the conjugate exponent $q$. Furthermore, the paper provides improved Landau-type theorems for these mappings. Under standard normalization, we derive explicit expressions for the radius of univalence $r_{0}$ and the radius of the inscribed schlicht disk $R_{0}$. The sharpness of these constants is discussed through the construction of extremal functions related to the Poisson kernel.