Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used $a,c\in \mathbb{R}$, for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper.

中文翻译：

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合流超几何函数的等价性的新条件

由于在许多特定情况下直接从函数的星形或凸性定义中检查条件可能很困难，因此在本文中，我们使用微分从属理论，尤其是可允许函数的方法来确定星形或凸性条件用于第一种融合的（Kummer）超几何函数。考虑到Miller和Mocanu在1990年使用的结果$\mathrm{一种}，C\in \mathbb{[R}$，对于融合（Kummer）超几何函数，在此研究中，使用a和c复数，并陈述了两个使被调查函数不合格的标准。为了说明论文原始结果的相关性，还包括一个示例。