On a class of pluriharmonic mappings in the unit polydisk

Abstract: In this paper, we introduce and study the class $\mathcal{W}<em>{\mathcal{H}_n<sup>0}(α)$ of normalized pluriharmonic mappings, characterized by a suitable bound on their second-order partial derivatives. We establish a one-to-one correspondence between this pluriharmonic class and an associated class of holomorphic functions, thereby extending a result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2019} to the setting of several complex variables. Furthermore, we obtain sharp coefficient bounds, growth estimates and a convex combination theorem for functions in $\mathcal{W}</em>{\mathcal{H}<em>n<sup>0}(α)$. Finally, we introduce sections (partial sums) of pluriharmonic mappings and investigate their properties for functions belonging to $\mathcal{W}</em>{\mathcal{H}_n<sup>0}(α)$.

• The paper introduces a new class of normalized pluriharmonic mappings, establishing a one-to-one correspondence with holomorphic functions.
• It derives sharp coefficient bounds and growth estimates using explicit multi-index calculations that extend classical harmonic mapping results.
• The study validates stability under convex combinations and partial sum approximations, offering robust tools for multidimensional complex analysis.

Summary of "On a class of pluriharmonic mappings in the unit polydisk" (2607.11629)

Overview and Motivation

The paper investigates the analytic and geometric properties of a new class $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$ of normalized pluriharmonic mappings in the unit polydisk of $\mathbb{C}^n$. These mappings are characterized by bounds on their second-order partial derivatives and extend classical results from harmonic mappings in one complex variable to pluriharmonic mappings in several complex variables. The study addresses explicit coefficient bounds, growth estimates, convexity properties, and partial sums (sections) for this functional class. The results generalize and complement prior work on harmonic mappings, especially the framework established in Ghosh and Vasudevarao (2019) for planar harmonic mappings.

Pluriharmonic Mappings: Structure and Fundamental Classes

Pluriharmonic mappings $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$ are expressed as $f = h + g$, with $h$ and $g$ holomorphic in the polydisk. Their normalization, derivative properties, and series expansions are carefully studied. Several geometric subclasses are defined:

• Starlike: $f(\mathbb{P}\Delta)$ is starlike with respect to the origin.
• Convex: $f(\mathbb{P}\Delta)$ is convex.
• Close-to-convex: $f(\mathbb{P}\Delta)$ is close-to-convex.

These classes, together with their normalized counterparts (where the co-holomorphic part vanishes to second order at the origin), form a hierarchy that reflects containment relations and geometric specialization. Notably, the subclass $\mathcal{S}^*_{\mathcal{H}_n}^0$ is a proper subset of both $\mathbb{C}^n$0 and $\mathbb{C}^n$1, echoing classical geometric function theory in higher dimensions.

Definition and Characterization of $\mathbb{C}^n$2

The central focus is on mappings $\mathbb{C}^n$3 satisfying:

$\mathbb{C}^n$4

for all $\mathbb{C}^n$5 and $\mathbb{C}^n$6 in the polydisk, with $\mathbb{C}^n$7.

A one-to-one correspondence is demonstrated: $\mathbb{C}^n$8 iff for any $\mathbb{C}^n$9 with $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$0, the holomorphic function $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$1 belongs to $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$2, i.e., it satisfies the analogous holomorphic differential inequality. This connection tightly integrates pluriharmonic and holomorphic theory, facilitating analytic reduction and transfer of results.

Sharp Coefficient Bounds

Explicit estimates are established for the coefficients in the expansions of $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$3 and $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$4:

• For a multi-index $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$5 with $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$6:

$f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$7

• Similar sharp bounds hold for $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$8 and $f : \mathbb{P}\Delta(0;1) \rightarrow \mathbb{C}$9 alone, with the same denominator structure. Extremality is demonstrated by explicit mapping constructions.

These results rigorously quantify the allowable magnitude of higher-order terms, providing a concrete foundation for distortion, growth, and geometric property analysis.

Growth Estimates

Growth estimates for $f = h + g$0 in $f = h + g$1 are derived:

$f = h + g$2

$f = h + g$3

Sharpness is again achieved for carefully constructed function examples. These bounds extend classical distortion theorems to pluriharmonic mappings in several complex variables, confirming the controlled expansion/contraction properties within the unit polydisk.

Structural Stability: Convex Combinations and Partial Sums

The class $f = h + g$4 is shown to be stable under convex combinations, i.e., any convex sum of mappings in the class remains within the class. This property ensures the robustness of geometric constraints under algebraic operations and multiparameter interpolation.

For partial sums (sections), using explicit truncations of the series expansions of $f = h + g$5 and $f = h + g$6, it is shown that $f = h + g$7, the section containing up to first degree terms from $f = h + g$8 and up to $f = h + g$9 from $h$0, remains in $h$1 for $h$2. This result provides insight into approximation of pluriharmonic mappings by finite polynomials, with implications for numerical and analytic applications in several complex variables.

Sufficient Conditions for Class Membership

A sufficient coefficient condition is provided:

$h$3

implies $h$4.

This quantitative restriction enables practical verification of class membership for candidate mappings, supporting both theoretical exploration and computational checks.

Implications and Future Directions

The results significantly advance geometric function theory in several complex variables, giving precise characterizations and explicit bounds for a broad functional class of pluriharmonic mappings. The extension from one variable harmonic mappings to pluriharmonic mappings in $h$5 opens new avenues for multidimensional complex analysis and its applications in geometry, mathematical physics, and PDEs.

These findings may serve as a foundation for further exploration of:

• Univalence, distortion, and covering properties for generalized harmonic and pluriharmonic mappings in high dimensions.
• Pseudoconvexity and potential theory in complex manifolds through the lens of pluriharmonic functions.
• Approximation methods for PDEs and geometric mapping problems based on partial sums and convex combinations.
• Further generalizations to domains beyond the unit polydisk, such as balls or more general Cartan domains.

The explicit coefficient and growth bounds are directly relevant for computational approaches, stability investigations, and extremal function construction in several complex variables.

Conclusion

This paper establishes a rigorous analytic and geometric framework for the class $h$6 of normalized pluriharmonic mappings in the unit polydisk. The authors derive a one-to-one correspondence with holomorphic mappings, prove sharp coefficient and growth bounds, validate convex combination stability, and analyze partial sums. These contributions extend classical harmonic mapping theory to the context of several complex variables, offering both theoretical depth and practical tools for ongoing research in geometric function theory and related fields.