Criteria for a certain class of the Carathéodory functions and their applications

Abstract

Abstract In this paper, we obtain some potentially useful conditions (or criteria) for the Carathéodory functions as a certain class of analytic functions by applying Nunokawa’s lemma. We also obtain several conditions for strong starlikeness and close-to-convexity as special cases of the main results presented here.

Keywords

• Carathéodory functions
• Differential subordination
• Strongly close-to-convex functions
• Starlike functions
• Strongly starlike functions