Sufficient Conditions for Caratheodory Functions

Abstract

In the present paper, we obtain several sufficient conditions for Carath\'eodory functions in the open unit disk $\mathbb{U}=\1\{ z\in\mathbb{C}:|z|<1 \2\}$. We also obtain sufficient conditions for $p$-valent or starlike functions. Moreover, we improve some results due to Nunokawa [Tsukuba J. Math., {\bf 13}(2) (1989), 453--455] as some special cases of main results.