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

TL;DR

This paper establishes sharp Bohr-type inequalities and Landau-type theorems for harmonic mappings in the unit disk with boundary functions in Lebesgue spaces, extending classical results and identifying optimal radii.

Contribution

It introduces new sharp Bohr radii for harmonic mappings with boundary functions in Lebesgue spaces and improves Landau-type theorems with explicit univalence and schlicht disk radii.

Findings

Derived sharp Bohr radius for harmonic mappings with $L^p$ boundary functions.

Extended classical Bohr inequalities to broader harmonic mapping classes.

Provided explicit radii for univalence and schlicht disks, with extremal functions demonstrating sharpness.

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.