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 $r \le (1-a)/(1-a+4/π)$, 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.

• The paper presents sharp Bohr radius estimates and improved Landau-type theorems for harmonic mappings defined by L^p boundary functions.
• It employs optimal coefficient bounds to quantify the influence of L^p regularity on univalence and schlicht disk inclusion.
• The results extend classical analytic findings, offering a robust framework for further exploration in geometric function theory.

Summary of "Bohr Radius and Landau-type Theorems for Harmonic Mappings with Boundary Functions in Lebesgue Spaces" (2604.14217)

Introduction

The paper addresses harmonic mappings in the unit disk $\mathbb{D}$ determined by boundary functions $F$ in Lebesgue spaces $L^p(\mathbb{T})$ ($1 \leq p \leq \infty$). Harmonic mappings are expressed as $f = h + \overline{g}$ with analytic $h$ and $g$, and boundary data is imposed via the Poisson integral. Two primary objectives are pursued: to establish sharp Bohr-type phenomena and to derive improved Landau-type theorems for classes of harmonic mappings induced by $L^p$-boundary functions, generalizing classical analytic results to broader function spaces.

Bohr Radius for Harmonic Mappings

Classical Bohr Radius and Sharp Estimates

The Bohr phenomenon concerns majorant series $$M_f(r) = |a_0| + \sum_{n=1}^\infty (|a_n| + |b_n|) r^n$$ for harmonic mappings bounded by $M$, searching for the maximal $F$0 so that $F$1. The paper establishes, via optimal coefficient bounds, the sharp Bohr radius:

$F$2

where $F$3 for $F$4. This result generalizes classical cases and is shown to be the best possible via extremal functions that saturate the bounds.

Extension to $F$5 Boundary Data

For harmonic mappings induced by boundary functions $F$6, sharp coefficient estimates are derived in terms of the conjugate exponent $F$7 ($F$8), leading to the Bohr radius:

$F$9

with $L^p(\mathbb{T})$0. In particular, $L^p(\mathbb{T})$1 yields $L^p(\mathbb{T})$2 and recovers the classical radius. These bounds are proven to be sharp, and the dependence on $L^p(\mathbb{T})$3 norms is precisely quantified.

Landau-type Theorems for Harmonic Mappings in $L^p(\mathbb{T})$4

Improved Univalence and Schlicht Disk Radii

Landau-type theorems ascertain maximal subdisks in which harmonic mappings are univalent, and guarantee that the image contains a schlicht (univalent) disk. Utilizing coefficient bounds parameterized by $L^p(\mathbb{T})$5 and $L^p(\mathbb{T})$6, explicit and improved formulas are provided:

• Univalence radius:

$L^p(\mathbb{T})$7

• Contained schlicht disk radius:

$L^p(\mathbb{T})$8

The derivation is based on detailed analysis of the expansion and sharp coefficient bounds, and shows that for $L^p(\mathbb{T})$9 the result recovers known optimal results for bounded harmonic mappings. While Bohr radii are achieved with extremal functions, the sharpness of Landau constants for general $1 \leq p \leq \infty$0 remains open, pending explicit construction of extremal boundary data.

Implications and Extensions

Practical and Theoretical Significance

The precise determination of Bohr radii for harmonic mappings with $1 \leq p \leq \infty$1 boundary functions allows geometric function theorists to quantify the influence of boundary regularity on controlled growth in the disk. The explicit dependence on $1 \leq p \leq \infty$2 and $1 \leq p \leq \infty$3 provides a flexible toolkit, applicable in analysis, PDE boundary problems, and conformal mapping theory.

Landau-type radius formulas lend themselves to use in geometric control of domains, regularity analysis for PDEs, and possible generalizations to quasiconformal, biharmonic, or pluriharmonic cases. The theoretical machinery developed here bridges classical analytic settings and the much broader context of Lebesgue space boundary data.

Future Directions

Several significant directions are proposed:

• Sharpness analysis: Determining whether the Landau constants $1 \leq p \leq \infty$4 and $1 \leq p \leq \infty$5 are optimal for $1 \leq p \leq \infty$6.
• Extremal construction: Characterizing boundary data in $1 \leq p \leq \infty$7 that attain extremal radii.
• Alternative normalizations: Extending results to mappings with Jacobian or dilatation normalization.
• Geometric boundary constraints: Incorporating further geometric properties, such as convexity or starlikeness, for boundary functions.

These directions underline the fundamental challenge of transitioning sharp classical results to generalized harmonic and geometric contexts.

Conclusion

This paper provides a comprehensive treatment of Bohr radius and Landau-type theorems for harmonic mappings with $1 \leq p \leq \infty$8 boundary functions, establishing sharp majorant series bounds and explicit radii of guaranteed univalence and schlicht disk inclusion. The results deliver technically strong generalizations of classical analytic theorems, quantify the boundary function’s influence, and lay a foundation for further developments in harmonic mapping theory and geometric function analysis. The open problems highlighted offer fertile ground for advancing the understanding of harmonic mappings in non-classical settings.