A Characterization of Concave Mappings Using the Carathéodory Class and Schwarzian Derivative

The purpose of this paper is to establish new characterizations of concave functions f defined in \({\mathbb {D}}\) $\mathbb{D}$ in terms of the operator \(1+zf''/f'\) $1+z{f}^{\prime \prime }/f^{\prime }$ , the Schwarzian derivative and the lower order. We will distinguish the cases when the omitted set is bounded or unbounded, and in the latter case, we will address the subclasses determined by the angle at infinity.