($α,β$)-Harmonic Functions: Theory & Estimates

• ($α,β$)-harmonic functions are C² functions on the unit disk satisfying a weighted Laplace-type equation that generalizes classical harmonic maps.
• They admit an absolutely convergent expansion using hypergeometric functions and yield sharp coefficient estimates via a generalized Heinz’s inequality.
• The theory connects boundary behavior, growth distortion, and univalence, extending classical results for harmonic and analytic mappings.

An $(\alpha, \beta)$-harmonic function is a $C^2$ function $u:\mathbb{D}\to\mathbb{C}$ on the unit disk $\mathbb{D}$ that satisfies a weighted Laplace-type equation $L_{\alpha,\beta}u=0$, where $L_{\alpha,\beta}$ is parametrized by real numbers $\alpha, \beta\notin \{-1,-2,-3,\dots\}$ with $\alpha + \beta > -1$. This generalizes classical harmonic and $\alpha$-harmonic functions by incorporating two real parameters, introducing additional flexibility and encompassing a broader class of mappings. Recent work establishes foundational properties of this class, including coefficient estimates, sharp inequalities, boundary behavior, and structural theorems, thus linking and generalizing classical extremal results for univalent sense-preserving harmonic maps of the disk (Qiao et al., 4 Dec 2025).

1. Definition and Series Expansion

Given parameters $$\alpha, \beta \in \mathbb{R} \setminus \{-1,-2,\dots\}$$ with $\alpha + \beta> -1$, the $(\alpha,\beta)$-harmonic operator is defined as: $$L_{\alpha,\beta} = (1-|z|^2)\left( (1-|z|^2)\partial_z\partial_{\bar z} + \alpha z \partial_z + \beta \bar z \partial_{\bar z} - \alpha \beta \right),$$ where $z \in \mathbb{D}$ and derivatives are with respect to complex coordinates.

A function $u$ is $(\alpha, \beta)$-harmonic if $L_{\alpha, \beta}u=0$ in $\mathbb{D}$. The normalization considered is that $u$ is sense-preserving ($|u_z| > |u_{\bar z}|$ everywhere), univalent, and maps $\mathbb{D}$ onto itself.

The general solution admits an absolutely convergent expansion: $$u(z) = \sum_{k=0}^{\infty} c_k F(-\alpha, k-\beta; k+1; |z|^2) z^k + \sum_{k=1}^{\infty} c_{-k} F(-\beta, k-\alpha; k+1; |z|^2) \bar{z}^k,$$ where $F(a,b;c;\cdot)$ denotes the Gauss hypergeometric function, and the convergence is ensured by $\limsup_{|k|\to\infty}|c_k|^{1/|k|} \leq 1$ (Qiao et al., 4 Dec 2025).

2. Sharp Coefficient Estimates: Heinz's Inequality

Heinz’s inequality is a classical result for univalent, sense-preserving harmonic self-maps of $\mathbb{D}$, providing a sharp lower bound involving leading coefficients. For $(\alpha, \beta)$-harmonic mappings, the generalized theorem asserts:

Let $u(z)$ be as above. Then the following sharp bound holds (Theorem 2.2): $$\boxed{ \left( \frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+\alpha)\Gamma(1+\beta)} \right)^2 \left( \frac{|c_{1}|^2}{(1+\alpha)^2} + \frac{3\sqrt{3}}{\pi} |c_0|^2 + \frac{|c_{-1}|^2}{(1+\beta)^2} \right) \geq \frac{27}{4\pi^2} }$$ As $\alpha,\beta\to 0$, this recovers the classical extremal case for harmonic self-maps, confirming the constant $\frac{27}{4\pi^2}$ is asymptotically sharp. Extremal equality is approached precisely in the classical case; no new extremals appear for general $(\alpha, \beta)$ (Qiao et al., 4 Dec 2025).

3. Proof Mechanisms and Boundary Behavior

The sharp coefficient inequality is obtained by adapting Hall’s approach for harmonic mappings to the $(\alpha, \beta)$ context. Essential elements are:

• Poisson-Integral Representation:

$$u(z) = \frac{1}{2\pi} \int_0^{2\pi} P_{\alpha, \beta}(ze^{-it}) u(e^{it})\,dt,$$

with $P_{\alpha, \beta}$ an explicit hypergeometric kernel. As $r\to 1^-$, the boundary function admits a Fourier-type expansion with coefficients determined by $\Gamma$-function ratios and the $c_k$.

• Application of Hall’s Argument Principle Estimate:

By focusing on the three leading Fourier terms, one relates boundary winding to coefficient structure, with univalence and the sense-preserving condition imposing the lower bound.

• Use of Hypergeometric Limit Formula:

The limit $F(a,b;c;1)$ is evaluated using

$$F(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},$$

ensuring the correct normalization for boundary terms (Qiao et al., 4 Dec 2025).

4. Special Cases and Corollaries

The general theory specializes to several important situations:

Case	Parameter Constraint	Consequence
Real-kernel	$\alpha = \beta > -1$	Recovers Heinz-type inequality of Long–Wang for $\alpha$-harmonic mappings.
Typically-real	All $c_k \in \mathbb{R}$	For $k\geq 2$: $$\left|\frac{\Gamma(1+\alpha) c_k}{\Gamma(k+1+\alpha)} - \frac{\Gamma(1+\beta) c_{-k}}{\Gamma(k+1+\beta)}\right| \leq \frac{1}{\Gamma(k)}\left|\frac{1}{1+\alpha} - \frac{c_{-1}}{1+\beta}\right|$$
Harmonic-starlike	$u\in \mathcal{S}_{(\alpha,\beta)}^0$	For $k\geq2$: $$|c_k| \leq \frac{(2k+1)(k+1)\Gamma(k+1+\alpha)}{6 k!\Gamma(2+\alpha)}$$, $$|c_{-k}| \leq \frac{(2k-1)(k-1)\Gamma(k+1+\beta)}{6(1+\alpha)k! \Gamma(1+\beta)}$$

Each of these results follows by combining the main coefficient machinery with classical extremal function estimates (Qiao et al., 4 Dec 2025).

5. Growth, Distortion, and Area Properties

$(\alpha,\beta)$-harmonic functions inherit and generalize many classical theorems for harmonic (and analytic) univalent maps, including:

• Radó’s Theorem and Koebe-type Covering Theorems: Adapted to the context of the weighted Laplace operator $L_{\alpha,\beta}$, guaranteeing boundary regularity and covering properties for the image domains.
• Area Theorem: The distortion and growth of $u$ are controlled in terms of its boundary function’s $L^p$ norm, connecting internal properties to the behavior on $|\partial\mathbb{D}|$.
• Coefficient Bound and Growth Estimate: For the subclass of $(\alpha,\beta)$-harmonic starlike functions $\mathcal{S}_{(\alpha,\beta)}^0$, all higher coefficients admit sharp explicit upper bounds as described above, interpolating between extremality and regularity in the function class (Qiao et al., 4 Dec 2025).

6. Connections to Classical Harmonic and Analytic Theory

The $(\alpha,\beta)$-harmonic family subsumes the classical case ($\alpha = \beta = 0$), automatically reproducing known results for harmonic mappings such as extremality of the harmonic Koebe function for Heinz’s inequality. The limiting behavior as $\alpha,\beta\to 0$ continuously interpolates these coefficient bounds and sharp constants, with no novel extremals arising for $\alpha + \beta > -1$.

Further, the method clarifies relationships between weighted boundary expansions (involving hypergeometric kernels) and classical Fourier analysis for univalent functions, while the coefficient growth condition ensures analogues of Carathéodory’s and Bieberbach’s theorems continue to govern the analytic structure in the generalized setting (Qiao et al., 4 Dec 2025).