Sufficient conditions for strongly Carathéodory functions

Recently, by applying the principle of subordination for analytic functions in the open unit disk UU, Kim and Cho [I. H. Kim, N. E. Cho, Sufficient conditions for Carathéodory functions, Comput. Math. Appl. 59 (2010), 2067–2073] considered several sufficient conditions for a family of Carathéodory functions. The main purpose of this paper is to investigate some (presumably new) sufficient conditions for the class of strongly Carathéodory functions in UU. One illustrative example and several corollaries of the main results presented here are also considered.