First-order Schwartian derivative and some classes of univalent functions

Abstract

In this paper we give estimate of $|S_f^{\prime }(0)|$, where $S_f(z)={\left(\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right)}^{\prime }-\frac{1}{2}{\left(\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right)}^2$ is the Schwartian derivative, and $f$ belongs to different classes of functions univalent in the open unit disc $\ID$.