Coefficient bounds and growth estimates for a class of pluriharmonic mappings in unit polydisk

Abstract: In this paper, we first introduce and study the class $\mathcal{P}<em>{\mathcal{H}_n<sup>0}(M)$ of normalized pluriharmonic mappings, characterized by a specific bound on the sum of their second-order partial derivatives. We prove a one-to-one correspondence between this pluriharmonic class and a class of holomorphic functions, extending the known result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2020} to the setting of several complex variables. Finally, we provide sharp coefficient bounds and growth estimates for functions in the class $\mathcal{P}</em>{\mathcal{H}_n<sup>0}(M)$.

• The paper extends classical geometric function theory to pluriharmonic mappings in several complex variables, establishing a one-to-one correspondence with holomorphic classes.
• The paper provides sharp coefficient estimates by deriving explicit bounds for the Taylor coefficients of pluriharmonic mappings in the unit polydisk.
• The study delivers tight growth estimates in the supremum norm, enabling precise analysis of mapping behavior in higher-dimensional complex domains.

Coefficient Bounds and Growth Estimates for Pluriharmonic Mappings in the Unit Polydisk

Introduction and Context

This work systematically extends classical geometric function theory from the harmonic and holomorphic settings in one complex variable to the pluriharmonic framework in several complex variables. The primary focus is on the unit polydisk $\mathbb{P}\Delta(0;1)\subset \mathbb{C}^n$ and the study of pluriharmonic mappings $f=h+g$, where $h$ and $g$ are holomorphic, normalized so that $f(0)=0$ and $\nabla f(0)=(1,\ldots,1)$.

Pluriharmonic functions, whose restriction to any complex line is harmonic, are central in higher-dimensional complex analysis. They play a vital role in Kähler geometry, potential theory, and complex geometric mapping theory, providing a natural multidimensional analogue to planar harmonic and holomorphic mappings.

The Class $\mathcal{P}_{\mathcal{H}_n^0}(M)$ and Its Properties

A principal contribution of this paper is the formal introduction of the class $\mathcal{P}_{\mathcal{H}_n^0}(M)$, defined for $M>0$ as the set of normalized pluriharmonic mappings

$$f = h + g \in \mathcal{H}_n^0: \sum_{l=1}^n \sum_{j=1}^n \sum_{k=1}^n \Re\left(z_l \frac{\partial^2 h(z)}{\partial z_j \partial z_k}\right) > -M + \sum_{l=1}^n \sum_{j=1}^n \sum_{k=1}^n \left|z_l \frac{\partial^2 g(z)}{\partial z_j \partial z_k}\right|$$

in the unit polydisk.

This definition generalizes the harmonic class $f=h+g$0 of Ghosh and Vasudevarao to several complex variables, offering a flexible and robust analytic framework for multidimensional geometric function theory.

A crucial structural result is a one-to-one correspondence: $f=h+g$1 if and only if $f=h+g$2 belongs to the holomorphic class $f=h+g$3 for each unit modulus $f=h+g$4. This result generalizes the planar theory to several variables and establishes a deep structural relationship between pluriharmonic and holomorphic coefficient- and growth-type inequalities in the context of multidimensional polydisks.

Sharp Coefficient Estimates

The paper provides explicit and sharp bounds on the Taylor coefficients of the co-holomorphic part $f=h+g$5 of functions in $f=h+g$6. If $f=h+g$7 with multi-index $f=h+g$8 and $f=h+g$9, then

$h$0

This bound is proven to be sharp; equality is achieved for extremal functions constructed in the paper. The result subsumes prior one-variable bounds and exhibits the correct combinatorial scaling in the multidimensional setting, with the multinomial coefficient accounting for the number of terms of homogeneous degree $h$1.

Furthermore, the authors derive inequalities for simultaneous combinations of $h$2 and $h$3, extending the real/imaginary and sum/difference coefficient bounds known from planar harmonic mappings.

Growth Estimates

For $h$4, the authors establish sharp global growth estimates in the supremum norm: $h$5

$h$6

for all $h$7. These are established by reduction to one-variable paths and careful combinatorial arguments, tracking the contributions of higher order terms to the maximal modulus on the polytorus.

Sharpness is again proved by constructing extremal functions whose coefficients saturate the bounds.

Analytical Implications

This analytic machinery substantially extends the coefficient theory for harmonic and holomorphic mappings to the pluriharmonic category in several complex variables. The explicit form of the bounds, with exact multinomial constants, allows precise discrimination of mapping behavior in higher dimensions, furthering the understanding of how non-holomorphic and nonlinear phenomena manifest in $h$8.

The connection to the holomorphic class via the mapping $h$9 is particularly powerful, reducing complicated pluriharmonic questions to holomorphic analogues, thereby enabling translation of a large corpus of classical results to the multidimensional context. In particular, these results may inform future developments in distortion theory, radius problems, and multidimensional versions of classical univalent function phenomena such as Bohr's phenomenon and covering theorems.

Theoretical and Applied Outlook

On the theoretical side, the work provides a template for further generalizations of harmonic analysis, Loewner-type theory, and coefficient inequalities in several complex variables. The explicit estimates will be directly applicable in the study of univalent, starlike, and convex pluriharmonic mappings, as well as boundary regularity and value distribution in higher-dimensional domains.

From an applied perspective, the precise growth and coefficient control for such mappings has potential relevance in models of multidimensional elasticity, field theories on Kähler and Hermitian manifolds, and areas where energy-minimizing deformations are relevant and the underlying geometric structure is inherently pluricomplex.

The extension framework given in the paper establishes an effective bridge between several complex variables, geometric function theory, and nonlinear analysis, and will likely stimulate subsequent work in higher-dimensional sharp inequalities, in both analytic and geometric settings.

Conclusion

The paper delivers a rigorous generalization of sharp coefficient and growth results for harmonic mappings from the unit disk to the setting of pluriharmonic mappings on the unit polydisk in $g$0. The explicit bounds achieved are tight, structurally nuanced, and intimately connected with the underlying holomorphic theory. These results not only extend the toolkit of geometric function theory in higher dimensions but also provide concrete pathways for further research in analytic, geometric, and applied aspects of several complex variables (2607.10616).