Certain polynomial subordinations (Q809219)

Let K denote the family of analytic functions f in the open unit disk \(\Delta\), normalized by \(f(0)=f'(0)-1=0\) and map \(\Delta\) onto a convex region. Theorem A. For a given integer \(n\geq 2\), let \(\lambda\) be a positive real number and \(\mu\) a complex number such that \(\lambda z+\mu z^ n\) is locally univalent in \(\Delta\). The subordinations \[ z/2\prec \lambda z+\mu a_ nz^ n\prec f(z)=z+\sum^{\infty}_{k=2}a_ kz^ k\quad (z\in \Delta)\text{ are valid for all } f\in K \] if and only if \(\lambda =1/2+(-1)^ n\mu\), \(0\leq (-1)^ n\leq 1/(2(n^ 2-1)).\) The particular case \(n=2\) of the above result was obtained by \textit{T. Basgöze}, \textit{J. L. Frank} and \textit{F. R. Keogh}, Can. J. Math. 22, 123-127 (1970; Zbl 0208.099). The necessary part of Theorem A was obtained by \textit{T. Chiba}, On convex univalent functions, Kenkyû Kiyô-Gakushûin Kotaka 7, 1-8 (1975).