On univalent functions with fixed second coefficient (Q789564)

Let \(f(z)=\{(\gamma +\alpha)^{-1}z^{1- \gamma}(z^{\gamma}F(z)^{\alpha})'\}^{1/\alpha}\) where \(\alpha\) is a positive real number, \(\gamma\) is a complex number such that \(\gamma +\alpha \neq 0\). The author claims to have solved the radii problem for f(z) when F(z) belongs to the univalent subclasses of the \(\lambda\)- spirallike functions of order \(\rho\), convex functions of order \(\rho\) or the class of functions F(z) where Re F'(z)\(>\rho\). The second coefficient in the Taylor expansion of F(z) is held constant. The results are sharp in particular when \(\gamma\) is a non-negative real number. Thus the author extends and generalizes some known results due to \textit{A. E. Livingston} [Proc. Am. Math. Soc. 17, 352-357 (1966; Zbl 0158.077)], the reviewer [Colloq. Math. 28, 133-139 (1973; Zbl 0238.30012)] and \textit{V. P. Gupta, P. K. Jain} and \textit{I. Ahmad} [Rend. Mat., VI. Ser. 12, 423- 430 (1979; Zbl 0437.30009)].