Certain subclasses of univalent functions with fixed second coefficient (Q1090794)

Let \(f_{\alpha}(z)=z+2\alpha z^ 2+c_ 3z^ 3+..\). denote a function that is analytic and univalent in the unit disk E with fixed second coefficient \(c_ 2=2\alpha\) \((0\leq \alpha <1)\). The class \(S^*(2\alpha)\) consists of all functions \(f_{\alpha}\) that are starlike with respect to \(w_ 0=0\), and the class U(\(\alpha)\) consists of all functions \(f_{\alpha}\) that are of bounded rotation. The author finds the range of values of the functionals \[ I(f_{\alpha})=Re(zf'_{\alpha}/f_{\alpha})+i Re(1+zf''_{\alpha}/f'_{\alpha})\quad and\quad I(f_{\alpha})=Re f'_{\alpha}+i Re(1+zf''/f'_{\alpha}) \] for \(f_{\alpha}\in S^*(2\alpha)\) and for \(f_{\alpha}\in U(\alpha)\) and fixed \(z\in E\). As a corollary, the author obtains the radius of convexity for the classes \(S^*(2\alpha)\) and U(\(\alpha)\). This article extends earlier work by the author.