Heinz's Inequality for (α,β)-Harmonic Functions

• The paper generalizes Heinz’s lemma by establishing an inequality for (α,β)-harmonic functions using gamma-function ratios and hypergeometric kernel distortions.
• The analysis utilizes series expansions and boundary behavior techniques to derive sharp coefficient bounds for normalized, sense-preserving univalent maps.
• The results underpin geometric applications such as area and covering theorems and recover classical bounds as (α,β) approaches (0,0).

Heinz's inequality for ($\alpha,\beta$)-harmonic functions is a precise coefficient estimate extending the classical Heinz’s lemma from harmonic self-maps of the unit disk to the larger family of ($\alpha,\beta$)-harmonic mappings. These are solutions of a generalized Laplace-type operator parametrized by real $\alpha,\beta$ satisfying $\alpha+\beta > -1$. The generalized inequality involves gamma-function ratios and accommodates hypergeometric kernel distortions intrinsic to the $(\alpha,\beta)$-harmonic framework, reducing to the classical result when $(\alpha,\beta)=(0,0)$. This extension provides sharp bounds on the lowest degree coefficients for normalized sense-preserving univalent $(\alpha,\beta)$-harmonic maps and underpins new developments in starlike subclasses and extremal coefficient growth, as well as geometric applications such as area and covering theorems (Qiao et al., 4 Dec 2025).

1. $(\alpha,\beta)$-Harmonic Mappings: Definition and Structure

A $C^2$ function $u:\mathbb{D}\to\mathbb{C}$ is $(\alpha,\beta)$-harmonic if

$$L_{\alpha,\beta}u(z) = (1 - |z|^2)\left[ (1 - |z|^2)\partial_z\partial_{\bar z} u + \alpha z\partial_z u + \beta \bar z\partial_{\bar z}u - \alpha\beta u \right] = 0$$

where $\alpha,\beta\in\mathbb{R}\setminus\{-1,-2,\dots\}$ and $\alpha+\beta>-1$. For $(\alpha,\beta)=(0,0)$, this reduces to the standard Laplacian, and $u$ is classical harmonic.

All $(\alpha,\beta)$-harmonic functions admit a canonical expansion involving hypergeometric series:

$$u(z) = \sum_{k=0}^\infty c_k F(-\alpha, k-\beta; k+1; |z|^2) z^k + \sum_{k=1}^\infty c_{-k} F(-\beta, k-\alpha; k+1; |z|^2) \overline{z}^k,$$

where $F$ denotes the Gauss hypergeometric function.

2. Statement of Heinz’s Inequality for $(\alpha,\beta)$-Harmonic Maps

For sense-preserving univalent $(\alpha,\beta)$-harmonic mappings $u:\mathbb{D}\to\mathbb{D}$, normalized ($c_1\ne0$, $|u(z)|<1$ on $\mathbb{D}$), the following coefficient inequality holds:

$$\Biggl( \frac{\Gamma(1+\alpha+\beta)}{\Gamma(1+\alpha)\,\Gamma(1+\beta)} \Biggr)^2 \Biggl( \frac{|c_1|^2}{(1+\alpha)^2} + \frac{3\sqrt{3}}{\pi} |c_0|^2 + \frac{|c_{-1}|^2}{(1+\beta)^2} \Biggr) \ge \frac{27}{4\pi^2}$$

The constant $27/(4\pi^2)$ is sharp. As $(\alpha,\beta)\to(0,0)$, the classical Heinz constant and standard harmonic results are recovered.

3. Analytical Framework: Expansions, Boundary Behavior, and Proof Strategy

The generalized inequality emerges from a combination of series expansion, explicit boundary analysis, and reduction to the classical harmonic case:

• Expanding $u$ on the unit circle ($z = e^{it}$ and $r\to1^-$) yields a Fourier series in which the coefficients $\tilde c_k$ involve hypergeometric and gamma prefactors:

$$\tilde c_k = c_k\,F(-\alpha,k-\beta;k+1;1) = c_k\,\frac{\Gamma(1+\alpha+\beta)\Gamma(k+1)}{\Gamma(1+\beta)\Gamma(k+1+\alpha)}$$

• Representing $u$ as $\tilde h(z) + \overline{ \tilde g(z) }$ on the boundary and using Hall’s method for the classical case, the inequality

$$|\tilde c_1|^2 + \frac{3\sqrt{3}}{\pi} |\tilde c_0|^2 + |\tilde c_{-1}|^2 \ge \frac{27}{4\pi^2}$$

is established.

• Substituting the expressions for $\tilde c_k$ in terms of $c_k$ and the hypergeometric/gamma factors yields the full $(\alpha,\beta)$-harmonic Heinz inequality.

4. Limiting Cases and Sharpness

The structure of the inequality and its parameters yields several notable specializations:

• Classical limit: As $(\alpha,\beta)\to(0,0)$, $F(-\alpha, k-\beta; k+1; 1)\to1$ and the gamma term tends to 1, recovering the standard sharp harmonic Heinz inequality.
• Real-kernel case: Setting $\alpha=\beta>-1$ specializes the result to the real-kernel $\alpha$-harmonic maps, as previously addressed in the literature. The same gamma-factor structure applies.
• Maps with vanishing $c_0$: For $u(0)=0$ ($c_0=0$), the inequality simplifies with the central term absent, without changing the lower bound.
• Sharpness: The lower bound is saturated by extremal normalized maps as in the harmonic case, and the introduction of hypergeometric scaling preserves this extremality.

5. Corollaries: Coefficient Estimates, Subclasses, and Geometric Consequences

Heinz’s inequality directly leads to additional results:

• Normalized case: For $u \in \SH_{(\alpha,\beta)}^0$ with $c_0=0$, $c_1=1$, $c_{-1}=0$, the inequality confirms the normalization is compatible with the lower bound, and facilitates explicit coefficient estimates for higher $c_k$ (see Conjecture 2.1 in (Qiao et al., 4 Dec 2025)).
• Starlike subclass: If $u$ is starlike, established coefficient bounds from the classical setting (Clunie–Sheil-Small) transfer to $(\alpha,\beta)$-harmonic functions via the same hypergeometric prefactors (Theorem 2.6).
• Univalence and geometric theorems: The inequality forms the basis for an $(\alpha,\beta)$-version of the Radó–Kneser–Choquet theorem (Theorem 2.5), Koebe-type covering theorems, and area theorems (Theorems 2.7 and 2.8), utilizing the coefficient control for geometric properties.

6. Significance in the Theory of Harmonic Mappings

The generalization of Heinz's inequality to $(\alpha,\beta)$-harmonic functions allows for sharp characterization of the distortion, growth, and coefficient bounds of a broad class of solutions to elliptic partial differential equations with variable kernels. The gamma-factor structure quantifies the precise inflation/deflation induced by the $(\alpha,\beta)$-deformation, anchoring a direct comparability with classical harmonic results and capturing additional phenomena originating from the underlying hypergeometric structure. This extension robustly supports the transfer and adaptation of classical geometric function theory results to the $(\alpha,\beta)$-harmonic context (Qiao et al., 4 Dec 2025).