Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes

Abstract

In this paper, we establish the sharp estimates of the pre-Schwarzian norm of functions $f$ in the Ma-Minda type starlike and convex classes $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$, respectively, whenever $\varphi(z)=3/\left(3+(\alpha-3)z-\alpha z^2\right)$ with $-3<\alpha\leq 1$, $\varphi(z)=(1+z)(1-s z)$ with $-1/3\leq s\leq 1/3$ and $\varphi(z)=1+z/\left((1-z) (1+\alpha z)\right)$ with $0\leq \alpha\leq 1/2$.