Pre-Schwarzian norm estimate for certain Ma-Minda Class of functions

Let ${\mathcal{S}}^{\ast }(\phi )$ be the class of all analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$, normalized by $f(0)=f^{\mathrm{\prime }}(0)-1=0$ that satisfy the subordination relation $z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\prec \phi (z)$, where $\phi$ is an analytic and univalent in $\mathbb{D}$ with $\mathrm{R}\mathrm{e}\phi (z)>0$ such that $\phi (\mathbb{D})$ is symmetric with respect to the real axis and stralike with respect to $1$. In the present article, we obtain the sharp estimates of the pre-Schwarzian norm of $f$ and the Alexander transformation $J[f]$ for functions $f(z)$ in the class ${\mathcal{S}}^{\ast }(\phi )$ when $\phi (z)=e^{\lambda z}$, $0<\lambda \le \pi \mathrm{/}2$ and $\phi (z)=\sqrt{1+cz}$, $0<c\le 1.$