Some distortion theorems for a class of convex functions (Q1066300)

Let A denote the class of analytic functions f in the unit disc \(E=\{z: | z| <1\}\) with \(f(0)=0\), \(f'(0)=1\), and \[ T=\{f\in A:| \frac{zf'(z)}{f(z)}-1| <1,\quad z\in E\}, \] \[ \tilde T=\{g\in A:| \frac{zg''(z)}{g'(z)}| <1,\quad z\in E\}. \] From the author's earlier paper [Proc. Am. Math. Soc. 87, 117-120 (1983; Zbl 0518.30015)] and the compacity of the class \(\tilde T,\) it follows that for \(g\in \tilde T\) \[ \sup_{| z| <1}| \frac{zg'(z)}{g(z)}- 1| =\rho <1 \] and therefore \(\tilde T\subset T\). In this paper the author obtains the precise value of \(\rho\) and some distortion theorems for the classes T and \(\tilde T.\)