Pre-Schwarzian norm estimate and characterization of certain harmonic mappings

Abstract: In this article, we consider certain class of harmonic mappings defined in the unit disk $\mathbb{D}={z\in\mathbb{C}: |z|<1}.$ Then we obtain pre-Schwarzian norm estimate of functions in the class. Next, we show that functions in the considered class are univalent and close-to-convex. Moreover, we discuss some growth and distortion theorems for associated analytic and co-analytic parts of harmonic mappings in the class. At last, we present coefficient estimate for the analytic part.

• The paper establishes a sharp pre-Schwarzian norm bound (≤ 3) for harmonic mappings based on the combined effects of analytic and co-analytic components.
• It proves that these mappings are univalent and close-to-convex by applying canonical analytic forms and strict coefficient as well as Bloch-type estimates.
• The study offers explicit distortion and growth bounds, bridging classical analytic function theory with novel extensions in harmonic mapping analysis.

Pre-Schwarzian Norm Estimates and Characterization of Harmonic Mappings

Introduction and Context

The paper "Pre-Schwarzian norm estimate and characterization of certain harmonic mappings" (2606.19015) addresses the geometric characterization and analytic norm estimates for a specialized class of harmonic mappings within the unit disk $\mathbb{D}$. Specifically, it investigates mappings $f = h + \overline{g}$ where $h$ is analytic, $g$ co-analytic, and the analytic part $h$ takes a canonical form with derivative $h'(z) = \frac{m}{m-z^m}$ for some integer $m$. The study leverages foundational notions from geometric function theory, including univalence, convexity, close-to-convexity, and the importance of pre-Schwarzian norm bounds for both analytic and harmonic functions.

Pre-Schwarzian Norm Estimates

The paper establishes a sharp upper bound for the pre-Schwarzian norm $\|P_f\|$ of harmonic mappings $f$ in the considered class $\mathcal{H}_R$: $f = h + \overline{g}$0 This result is achieved by combining the analytic norm for the canonical $f = h + \overline{g}$1 ($f = h + \overline{g}$2) with the contribution from the dilatation $f = h + \overline{g}$3, leveraging Schwarz-Pick bounds to show that the distortion due to the harmonic part adds at most 1 to the analytic norm. The paper demonstrates sharpness of this bound via explicit extremal functions, notably for $f = h + \overline{g}$4 and dilatation $f = h + \overline{g}$5 as $f = h + \overline{g}$6.

This norm estimate is significant as it ensures uniform local univalence and provides global geometric control over the harmonic mappings. The explicit norm calculation provides a constructive analytic framework for further distortion and growth estimates.

Univalence, Convexity, and Close-to-Convexity

The paper proves that every $f = h + \overline{g}$7 is univalent and close-to-convex in $f = h + \overline{g}$8. The proof utilizes the convexity of the analytic part $f = h + \overline{g}$9, relying on the characterization

$h$0

which is shown to hold for the parameterized family $h$1, including $h$2 and $h$3 cases. The reduction of the harmonic mapping's geometric properties to those of $h$4 exploits the structure results of Clunie and Sheil-Small.

This guarantees that mappings in $h$5 are not just locally injective but globally well behaved in their image domains, with close-to-convexity ensuring no interior self-intersections and supporting further analytic norm and distortion analyses.

Bloch-Type Estimates and Distortion Theorems

For harmonic mappings $h$6, the paper verifies Bloch-type norm bounds: $h$7 This control confirms that $h$8 is a harmonic Bloch mapping, with the supremum attained for maximal $h$9 approaching the boundary. The analytic framework provides further distortion estimates for $g$0 and $g$1: $g$2 for $g$3, establishing both lower and upper bounds that are sharp. Growth theorems for $g$4 are also provided, parameterized by the dilatation's value at the origin, with explicit integral bounds.

These results provide quantitative insight into the boundary behavior, ensuring no super-Bloch growth and capping the distortion introduced by both analytic and co-analytic parts within $g$5.

Coefficient Bounds and Implications

The paper further delivers sharp coefficient bounds for the analytic part: $g$6 valid for all $g$7, with maximality achieved for $g$8. This aligns with classical Bieberbach-type estimates for normalized univalent functions, showing that the harmonic extension via the studied class does not inflate analytic distortion coefficients beyond classical bounds.

Coefficient estimates are critical for controlling function behavior in the unit disk, especially for the expansion-based analytic manipulation and in establishing further extremal geometric results.

Practical and Theoretical Implications

From a theoretical perspective, the work provides sharp analytic norm and coefficient estimates for a rich subclass of harmonic mappings, bridging classical analytic function theory with the harmonic mapping extension and offering global geometric insights (univalence, close-to-convexity, distortion). Practically, these results support controlled image domain manipulation in conformal and quasiconformal mapping contexts—relevant to mathematical modeling, geometric function theory, and applications in complex analysis.

The precise bounds for pre-Schwarzian and Bloch norms, along with distortion theorems, create a foundation for further investigation into nonlinear mapping classes, extremal problems, and the integration of analytic and harmonic mapping geometries. This can motivate future extensions to more general dilatation forms, higher-order norm estimates, and stability analyses for mappings beyond the unit disk.

Conclusion

The paper systematizes pre-Schwarzian norm estimation and geometric characterization for a distinguished class of harmonic mappings with analytic part $g$9, providing sharp norm bounds ($h$0), univalence, close-to-convexity, Bloch-type bounds, distortion/growth theorems, and coefficient estimates. These results enhance the structural understanding of harmonic mappings in geometric function theory, with implications for analytic extension, geometric distortion control, and further harmonic mapping research avenues.