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

Abstract Let S * ( φ ) ${\mathcal{S}}^{\ast }(\varphi )$ be the class of all analytic functions f in the unit disk D = { z ∈ C : | z | < 1 } $\mathbb{D}=\left\{z\in \mathbb{C}\,:\,\vert z\vert < 1\right\}$ , normalized by f (0) = f ′(0) − 1 = 0, that satisfy the subordination relation zf ′( z )/ f ( z ) ≺ φ ( z ), where φ is analytic and univalent in D $\mathbb{D}$ with Re φ ( z ) > 0. Furthermore, assume φ ( D ) $\varphi (\mathbb{D})$ is symmetric with respect to the real axis and starlike with respect to 1. In the present article, we obtain sharp estimates for the pre-Schwarzian norm of f and the Alexander transformation J [ f ] for functions f ( z ) in the class S * ( φ ) ${\mathcal{S}}^{\ast }(\varphi )$ when φ ( z ) = e λz (0 < λ ≤ π /2) and φ ( z ) = 1 + c z $\varphi (z)=\sqrt{1+cz}$ (0 < c ≤ 1).