arXiv 2501.016632025-01math.CV

由微分不等式定义的一类调和映射的几何分析

Geometric analysis of a class of harmonic mappings defined by a differential inequality

Indian Institute of Technology Bhubaneswar（印度布巴内斯瓦尔理工学院）；Raiganj University（拉甘杰大学）

Vasudevarao Allu, Raju Biswas, Rajib Mandal

备注：: 10 Pages, 2 figures and 1 table

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AI总结：

本文研究了由微分不等式定义的调和映射类，推导了系数界、增长界和单值性半径，并展示了该类在凸组合下的封闭性，扩展了调和映射理论中的已知结果。

摘要：

本文介绍了并系统研究了由微分不等式定义的归一化调和映射类$\mathcal{P}_{\mathcal{H}}^{0}(α,M)$，该类映射$f = h + \overline{g}$定义在单位圆盘$\mathbb{D}$上，满足不等式$\text{Re}\left((1-α)h'(z) + αz h''(z)\right) > -M + \left|(1-α)g'(z) + αz g''(z)\right|$，其中$M > 0$，$α\in (0,1]$，且$g'(0) = 0$。该类扩展了具有正实部函数的调和类，并提供了一个统一的分析框架。我们获得了分析部分和共分析部分的尖锐系数界，建立了尖锐增长界，并确定了单值性、星形性和凸性半径。此外，我们证明了$\mathcal{P}_{\mathcal{H}}^{0}(α,M)$在凸组合下封闭，并在参数适当限制下也封闭于卷积下。我们的发现扩展并推广了调和映射理论中的已知结果。

英文摘要：

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality \[ \text{Re}\left((1-α)h'(z) + αz h''(z)\right) > -M + \left|(1-α)g'(z) + αz g''(z)\right|\quad\text{for}\quad z\in\Bbb{D}, \] where $M > 0$, $α\in (0,1]$, and $g'(0) = 0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.

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