New Conditions for Univalence of Confluent Hypergeometric Function

• Main Author: Georgia Irina Oros
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Series: Symmetry
• Subjects: differential subordination; analytic function; starlike function; convex function; univalent function; dominant
• Online Access: https://www.mdpi.com/2073-8994/13/1/82

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 <inline-formula><math display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, for the confluent (Kummer) hypergeometric function, in this investigation <i>a</i> and <i>c</i> 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.