A note on subordination (Q1079977)

Let A denote the class of functions of the form \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) which are analytic in the unit disc in \({\mathbb{C}}\). Let S be the subclass of A consisting of univalent functions. Then a function f of S is said to be starlike of order \(\alpha\) if and only if \(Re\{zf'(z)/f(z)\}>\alpha\) for some \(\alpha\) \((0\leq \alpha <1)\). Further a function f of S is said to be convex of order \(\alpha\) if and only if \(Re\{zf''(z)/f(z)\}>\alpha\) for some \(\alpha\) \((0\leq \alpha <1).\) The author uses a result of \textit{T. J. Suffridge} [Duke Math. J. 37, 775- 777 (1970; Zbl 0206.362)] to prove that if f is convex of order \(\alpha\) then the image of \(D_ r=\{z:\) \(| z| \leq r\}\) \((0<r<1)\) under f' is contained in the image of \(D_ r\) under \(e^{4(\alpha -1)/(1-z)}\). The author uses the same method to prove that if f is starlike of order \(\alpha\) then the image of \(D_ r\) under f(z)/z is contained in the image \(D_ r\) under \(e^{4(1-\alpha)/(1-z)}\). These results are sharp.