An operator on families of univalent functions
Let S denote the class of functions \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) analytic and univalent in the unit disk \(\Delta =\{z:| z| <1\}\), and let \(S^*(\alpha)\) (0\(\leq \alpha \leq 1)\) denote the subclass of S of functions f which satisfy Re(zf'(z)/f(z))\(\geq \alpha\) (z\(\in \Delta)\). Furthermore, \(S^*_ 1(\alpha)\) is the subclass of \(S^*(\alpha)\) consisting of functions f for which \[ | (zf'(z)/f(z))-1| \leq 1-\alpha \quad (z\in \Delta). \] It is known that for f to be in \(S^*_ 1(\alpha)\) it is sufficient that \[ \sum^{\infty}_{n=2}(n-\alpha)| a_ n| \leq 1-\alpha \] and that this condition is necessary if f is of the form \[ (1)\quad f(z)=z- \sum^{\infty}_{n=2}a_ nz^ n,\quad a_ n\geq 0. \] The subclass of \(S^*(\alpha)\) consisting of functions f of the form (1) is denoted by \(T^*(\alpha)\), and \(T^*(\alpha)\subset S^*_ 1(\alpha)\). Also, \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) (z\(\in \Delta)\) is said to be in K, the class of convex functions, if \(Re(1+zf''(z)/f'(z))\geq 0\) for z in \(\Delta\). In this note the author investigates relationships between f and g: \[ g(z)=g_{\gamma}(z) = z(f(z)/z)^{\gamma}\quad (\gamma \text{ real}) \] when f is in S or in one of the above subclasses. He determines sufficient conditions in the form of coefficient inequalities for g to be in \(S^*(\alpha)\), \(S^*_ 1(\alpha)\) or K, and also obtains necessary and sufficient conditions on f for g to be in \(S^*(\alpha)\), \(S^*_ 1(\alpha)\) or K from which coefficient bounds on g are derived. In addition, he shows that \(f\in S\) implies \(g\in S\) only for \(\gamma =0,1\) and a coefficient inequality for g when \(f\in T^*(\alpha)\) is also given.
scientific article; zbMATH DE number 3998147