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

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 $1\le p\le \mathrm{\infty }$. We first establish a sharp Bohr-type inequality for the class of bounded harmonic mappings. Specifically, we prove that for a fixed analytic part $\mathrm{\mid }{a}_0\mathrm{\mid }=aM$, the majorant series $M_f(r)$ satisfies $M_f(r)\le M$ for $r\le (1-a)\mathrm{/}(1-a+4\mathrm{/}\pi )$, 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 $r_p=1\mathrm{/}(2C_q+1)$, 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.