On functions with negative coefficients which are starlike and convex with respect to \(n\)-symmetric points (Q1272000)

The author studies the classes \(\overline{S}_n^*(a,b)\) and \(\overline{S}_n^0(a,b)\) of regular functions \(f(z)\) defined in \(E=\{z:\;| z| <1\}\) and such that \[ f(z)=a_1 z - \sum_{k=2}^{\infty} a_k z^k,\;\;a_1>0,\;a_k\geq 0; \] moreover, \[ \frac{zf'(z)}{f_n(z)} \prec \frac{1+az}{1+bz} \] for the functions \(f\in \overline{S}_n^*(a,b)\) and \[ \frac{(zf'(z))'}{f_n'(z)} \prec \frac{1+az}{1+bz} \] for the functions \(f\in \overline{S}_n^0(a,b)\), where \[ f_n(z)=\frac{1}{n}\sum_{m=0}^{n-1}\varepsilon^{-m}f(\varepsilon^m z),\;\;\varepsilon=\exp(2\pi i/n),\;n=1,2,\ldots . \] The real numbers \(a\) and \(b\) meet the inequalities \(-1\leq b<0\) and \(b<a\leq 1\). The main results of the article can be described as follows. Necessary and sufficient conditions for an \(f\) to be in one of these classes are established. Boundary points of the closures of convex hulls of these classes are determined. A covering theorem for the functions in \(\overline{S}_n^*(a,b)\) is obtained. Distortion theorems are presented which give sharp estimates for functions \(f,f'\) (\(f\) belongs to one of these classes). It is shown that the class \(\overline{S}_n^*(a,b)\) is invariant under the transformation \(F(z)=(c+1)z^{-c}\int_{0}^{z}t^{c-1}f(t) dt\) to within the numbers \(a,b\) (\(a,b\to a',b'\) under the transformation), where \(c\) is a nonnegative integer.