An estimation of the pre-Schwarzian norm for certain classes of analytic functions

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions $f$ in the class ${\mathcal{S}}^{\ast }(\phi )$ and $\mathcal{C}(\phi )$ when $\phi (z)=1\mathrm{/}(1-z{)}^s$ with $0<s\le 1$ and $\phi (z)=(1+sz{)}^2$ with $0<s\le 1\mathrm{/}\sqrt{2}$, where ${\mathcal{S}}^{\ast }(\phi )$ and $\mathcal{C}(\phi )$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $\phi$, respectively.