Some properties of certain multivalent functions (Q5905404)

Applying a lemma due to \textit{I. S. Jack} [J. Lond. Math. Soc., II. Ser. 3, 469-474 (1971; Zbl 0224.30026)], the author derives some results for functions of the form \(f(z)=z^ p+a_{p+1}z^{p+1}+\dots\) which are regular in the unit disk \(U=\{z:\;| z|<1\}\) and satisfy there the condition \(| f^{(p)}(z)-p!|<p!-\alpha\), where \(p\) is a natural number and \(0\leq\alpha<p!\). For example, he proves that \(\hbox {Re}\{e^{i\beta} f^{(p-1)}(z)/z\}>0\) for \(z\) in \(U\) and \(|\beta|\leq{\pi \over 2}-\arcsin{p!-\alpha \over p!}\), \(\beta\) real.