(α,β)-Harmonic Mappings Overview

• (α,β)-harmonic mappings are generalized harmonic functions that solve specialized elliptic PDEs, extending classical harmonicity with parameter-driven uniqueness in Dirichlet problems.
• They include log-harmonic and univalent subclasses where geometric constraints enforce sense-preservation and yield explicit coefficient and growth bounds.
• Analytic techniques such as Poisson kernels, hypergeometric expansions, and convolution methods provide precise derivative estimates, boundary behavior control, and stability results.

A function $u:\D\to\C$ (where $\D$ is the unit disk) or, more generally, a map in analytic/harmonic function theory, is called $(\alpha,\beta)$–harmonic if it solves a certain elliptic PDE parameterized by $(\alpha,\beta)\in\C^2$ (often real values are taken in applications), generalizing classical harmonicity. There exist several major families of $(\alpha,\beta)$–harmonic mappings, each motivated by distinct operator-theoretic, geometric, or univalent function-theoretic frameworks. The main types include: (i) solutions to $(\alpha,\beta)$–Poisson or Laplacian-type elliptic PDEs (Sobolev/Hardy and extension theory), (ii) subclasses of sense-preserving harmonic mappings with $\alpha,\beta$ as geometric constraint parameters, and (iii) $(\alpha,\beta)$–log-harmonic mappings with explicit factor structures. These classes have led to sharp function-theoretic results such as coefficient estimates, extremal growth bounds, convolution theorems, higher dimensional analogues, and derivative distortion inequalities.

1. Principal Definitions and Operator Framework

Several non-equivalent, but interrelated definitions of $(\alpha,\beta)$–harmonic mappings are foundational:

(a) Classical $(\alpha,\beta)$–harmonic operator (Elliptic PDE context)

For $\alpha,\beta\in\C\setminus\{-1,-2,\dots\}$ with $\Re\alpha+\Re\beta>-1$, define

$$A_{\alpha,\beta}u = (1-|z|^2)\,\partial_z \partial_{\overline{z}}u + \alpha\,\partial_z u + \beta\,\partial_{\overline{z}}u - \alpha\beta\,u.$$

A $C^2$ function $u:\D\to\C$ is $(\alpha,\beta)$–harmonic if $A_{\alpha,\beta}u=0$ in $\D$ (Khalfallah et al., 2023, Arsenović et al., 2023). The unique solution to the Dirichlet problem with boundary data $f\in C(\mathbb{T})$ (circle) is

$$u(z) = P_{\alpha,\beta}[f](z) := \int_{0}^{2\pi} P_{\alpha,\beta}(z,e^{i\theta})\,f(e^{i\theta})\,\frac{d\theta}{2\pi}$$

where

$$P_{\alpha,\beta}(z,e^{i\theta})=C_{\alpha,\beta}\frac{(1-|z|^2)^{\alpha+\beta+1}} {(1-ze^{-i\theta})^{\alpha+1}(1-\overline{z}e^{i\theta})^{\beta+1}}$$

and $$C_{\alpha,\beta} = \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}$$ (Khalfallah et al., 2023, Arsenović et al., 2023).

(b) $(\alpha,\beta)$–log-harmonic mappings

A mapping $f:\D\to\C$ (with simple zero at the origin, $f(0)=0$) is log-harmonic if it can be written as

$$f(z)=z|z|^{2\beta}h(z)\overline{g(z)},\quad \mathrm{Re}\,\beta>-\tfrac12,$$

where $h,g$ are analytic in $\D$, $h(0)\neq0$, $g(0)=1$; with a certain analytic “second complex dilatation” $w(z)$, $|w(z)|<1$, enforcing sense-preservation (Kargar et al., 2017).

(c) $(\alpha,\beta)$–harmonic subclasses in univalent function theory

For harmonic mappings $f=h+\overline{g}$ in $\D$ (with standard normalization), various $(\alpha,\beta)$–classes arise, e.g.,

• $\mathcal B_{\mathcal H^0}(\alpha,\beta)$: functions so that $|zh''+\alpha(h'-1)| \leq \beta - |zg''+\alpha g'|$ (Mathi et al., 2021).
• $\mathcal P_H^0(\alpha), \mathcal G_H^0(\beta)$: classes imposing Re$\,h'-\alpha>|g'|$ and Re$\,h(z)/z-\beta>|g(z)/z|$ respectively (Li et al., 2015).

These subclasses yield geometric and combinatorial structure (close-to-convexity, univalence, convolution closure).

2. Analytic Structure, Poisson Kernels, and PDE Theory

The $(\alpha,\beta)$–harmonic operator is elliptic-degenerate (unless $\alpha+\beta+1= 0$), and its null-solutions generalize classical harmonic, $a$–harmonic, and weighted Laplacians. The Poisson kernel $P_{\alpha,\beta}(z,\zeta)$ provides explicit integral formulas for all boundary regularities above $L^1(\mathbb{T})$ (Khalfallah et al., 2023, Arsenović et al., 2023, Gajic et al., 2023). These kernels are positive, smooth in the disk, and parameterized with gamma-functions, and they admit product structure in the polydisc for separately $(\alpha,\beta)$–harmonic functions: $$P_{\alpha,\beta}(z,\zeta) = \prod_{j=1}^n P_{\alpha_j,\beta_j}(z_j,\zeta_j)$$ for $z\in\D^n$ and $\zeta\in\mathbb{T}^n$.

A key feature is the homogeneous (hypergeometric-function) expansion for these solutions: $$u(z) = \sum_{k\ge0}c_k\, F(-\alpha,k-\beta ; k+1 ; |z|^2) z^k + \sum_{k\ge1}c_{-k}\, F(-\beta,k-\alpha ; k+1 ; |z|^2)\overline{z}^k,$$ providing precise control over regularity and boundary behavior (Khalfallah et al., 2023, Gajic et al., 2023).

3. Geometric Properties, Subordination, and Close-to-Convexity

$(\alpha,\beta)$–harmonic classes reveal rich geometric phenomena:

• For $\mathcal{B}_{\mathcal{H}^0}(\alpha,\beta)$, if $0<\beta\leq 1+\alpha$, every element is close-to-convex, and hence univalent, a result established via reduction to an analytic subclass and application of the Clunie–Sheil-Small lemma (Mathi et al., 2021).
• Coefficient bounds: $|a_n|,|b_n|\leq \beta/(n(n+\alpha-1))$.
• Growth: $|f(z)|$ lies between $|z| - \frac{\beta|z|^2}{2(1+\alpha)}$ and $|z| + \frac{\beta|z|^2}{2(1+\alpha)}$.
• Closure under convex combinations and convolution with convex analytic maps is established via Hadamard product machinery and subordination principles.

In the log-harmonic context, starlikeness of order $\alpha$ is enforced via

$$\mathrm{Re}\left\{ \frac{z f_z - \overline{z}f_{\overline{z}}}{f(z)} \right\} > \alpha,$$

with subordination playing a critical role: $(z h'/h - z g'/g) \prec \frac{2(1-\alpha)z}{1-z}.$ This enables explicit integral and coefficient representations for starlike log-harmonic mappings (Kargar et al., 2017).

4. Derivative Estimates, Hardy Spaces, and Schwarz–Pick Theory

The analysis of function-theoretic and mapping-theoretic properties relies on sharp $L^p$–norm and derivative estimates:

• For $(\alpha,\beta)$–harmonic Poisson extensions with $f\in L^p(\mathbb{T})$, all first derivatives $\partial_z u$, $\partial_{\overline{z}}u$ belong to generalized Hardy space $H_G^p(\D)$ when $\alpha+\beta>0$, with explicit two-sided estimates. If $\alpha+\beta<0$, $H_G^1$ boundedness forces triviality or polyharmonic (finite Fourier) structure (Khalfallah et al., 2023).
• Schwarz–Pick-type theorems are established for all $(\alpha,\beta)$ satisfying $\Re\alpha+\Re\beta>-1$: for $u=P_{\alpha,\beta}[f]$,

$$|Du(z)| \leq C_{\alpha,\beta,p}\, \frac{\|f\|_{L^p(\mathbb{T})}}{(1-|z|^2)^{1+2/p}}$$

with sharp constants at the origin involving elliptic integrals, and similarly for higher order derivatives; every such derivative picks up an extra $(1-|z|^2)^{-1}$ singularity (Arsenović et al., 2023).

5. Convolution Theorems, Polydisc Extensions, and Section Theory

Harmonic convolution (Hadamard product) techniques produce stable subclasses:

• For $f\in\mathcal{P}_H^0(\alpha)$, $F\in\mathcal{G}_H^0(\beta)$, their convolution $f*F$ is close-to-convex in $\D$ when $1-2(1-\alpha)(1-\beta)\geq0$ (Li et al., 2015).
• Diagonal sections $s_{n,n}(f)$ enjoy explicit univalence/convexity radii, generalizing Szegő’s theorem: $s_{n,n}(f)$ is univalent and close-to-convex in $|z|<1/4$ for $n\geq2$ and convex unless $n=3$ (Li et al., 2015).

For separately $(\alpha,\beta)$–harmonic mappings on the polydisc $\D^n$, analogous Poisson kernel structure grants unique solutions to the Dirichlet problem, full $H^p$–theory (boundary convergence, integral representation, maximal function inequalities), and hypergeometric-homogeneous expansions (Gajic et al., 2023).

6. Impact of Parameters $(\alpha,\beta)$ and Critical Phenomena

The roles of $\alpha$ and $\beta$ are crucial:

• In PDE-type $(\alpha,\beta)$–harmonicity, the requirement $\alpha+\beta>-1$ is central to the kernel’s integrability and to uniqueness of solutions (Khalfallah et al., 2023).
• In log-harmonic or geometric contexts, larger $\alpha$ or $\beta$ adjust the “angular defect” or dilatation control, respectively. For $\alpha>1/2$, additional sharp lower bounds are imposed (e.g., $\mathrm{Re}\,h/g>1/(3-2\alpha)$) (Kargar et al., 2017).
• As $|\mathrm{Re}\,\beta|\to1/2$ or $\alpha+\beta\to -1$, regularity properties degenerate: Hardy space inclusion fails, extremal growth becomes singular, and rigidity theorems confine the possible function class to trivial or polynomial cases (Khalfallah et al., 2023, Arsenović et al., 2023).

Parameter Range	Mapping Behavior	Key Structural Consequences
$\alpha+\beta>0$	Derivatives in all $H_G^p(\D)$, full Hardy/Boundedness	Unique integral Poisson extensions, boundary norm convergence
$\alpha+\beta=0$	Rigidity: only trivial solutions if derivative in $H_G^1$	Only the zero function unless extra polynomial structure present
$\alpha+\beta<-1$	No Poisson kernel: non-integrability	No nontrivial continuous solutions for general $f$
$\mathrm{Re}\,\beta\to -1/2$	Dilatation constraint tightens; Jacobian bounds degenerate	Requires stronger coefficient (majorization) conditions

7. Recent Advances and Open Problems

Recent work elucidates precise Schwarz–Pick, coefficient, and distortion inequalities for $(\alpha,\beta)$–harmonic maps, and extends the theory to higher dimensions, maximal function estimates, and convolution closure (Gajic et al., 2023, Arsenović et al., 2023, Mathi et al., 2021). Current challenges include:

• Identification of extremal boundary data attaining equality in derivative estimates for all ranges of $\alpha, \beta$.
• Extension of rigidity and boundary regularity theory to more general domains and weights.
• Understanding the full geometric mapping picture (e.g., convexity, starlikeness) for generalized log-harmonic and close-to-convex $(\alpha,\beta)$–maps beyond weighted/unit disk settings.
• Detailed study of the transition regimes at $\alpha+\beta=0$ and at the limits of allowed parameter sets.

Major open directions involve the interplay between analytic, geometric, and operator-theoretic properties as parameter values approach critical thresholds, and further generalizations to mappings on complex manifolds or with more intricate weight structures.