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

Let S * ( φ ) S(φ ) be the class of all analytic functions f in the unit disk D = { z ∈ C : | z | < 1 } D=∈ C\,:\, z < 1\ , 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 D with Re φ ( z ) > 0. Furthermore, assume φ ( D ) φ (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 ) S(φ ) when φ ( z ) = e λz (0 < λ ≤ π /2) and φ ( z ) = 1 + c z φ (z)=√1+cz (0 < c ≤ 1).