发表机构
Babasaheb Bhimrao Ambedkar University（巴巴沙海布·比姆拉奥·安贝德卡尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为调和映射及其族建立了$\varphi$-正规性的Lappan型准则，涉及高阶球面导数与测试值数量，并推广至多项式情形。
AI 中文摘要
本文给出了单位圆盘$\mathbb{D}$中保向调和映射及其族的$\varphi$-正规性的Lappan五点型准则。我们建立了一个涉及$k$阶高阶球面导数以及$k+2$个不同测试值的正规性准则，在适当条件下成立。我们进一步在涉及解析部分和共轭解析部分的附加微分条件下，将测试值的数量减少到$\left[\frac{k}{2}\right]+2$个。我们还证明了一个推广，其中球面导数被一个具有非负实系数的多项式所取代。
英文摘要
In this paper, we present Lappan's five-point type criteria for $φ$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under suitable conditions. We further reduce the number of test values to $\left[\frac{k}{2}\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.
