Some radius of convexity problems for certain classes of analytic functions (Q1065193)

A class of functions \(T_ k\) is introduced. \(f\in T_ k\) if and only if \(f(0)=0=1-f(0)\) and there exists a function g of bounded boundary rotation (this class denoted by \(V_ k)\) such that \[ Re(f'(z)/g'(z))>0\quad | z| <1. \] Let f be analytic such that f'/g' belongs to a certain class of functions with positive real part where \(g\in V_ k\) or \(T_ k\). Then an upper bound for the radius of convexity of f is obtained. Similarly if \[ f_{\alpha}(z)=\int^{z}_{0}(f'(t))^{\alpha}dt \] where \(f\in V_ k\) or \(T_ k\), then the authors find an r such that f maps \(| z| <r\) onto a domain which is convex or close-to-convex, respectively.