Differential subordination for bounded turning functions using pre-Schwarzian and the Schwarzian derivatives

A normalized analytic function defined on the open unit disk is a bounded turning function if its derivative has positive real part. Such functions are univalent, and therefore, we find sufficient conditions for a function to be a bounded turning function. In this paper, we prove a general differential subordination theorem in terms of the derivative, the pre-Schwarzian derivative, and the Schwarzian derivative, providing sufficient conditions for a function to be a bounded turning function. We then apply the result to obtain several simple sufficient conditions.