On the coefficients of small univalent functions (Q1384127)

In this paper, a univalent function \(f(z)= \sum^\infty_{n=0} a_nz^n\) is small if \(\alpha>0\) is small in the inequality \((1-| z|^2) | f''(z)/f'(z) |\leq \alpha\) for \(| z | <1\). The authors prove the following theorem. For sufficiently small \(\alpha>0\), there exists a univalent function \(f\) with \((1-| z|^2) | f''(z)/f'(z) |\leq \alpha\) for \(| z | <1\) such that, for some constant \(c>0\), the inequality \(| a_n|> n^{c\alpha^2-1}\) holds for infinitely many \(n\). The proof depends on results by \textit{N. G. Makarov} [Proc. Am. Math. Soc. 96, 233-236 (1986; Zbl 0623.30025)] and by \textit{L. Carleson} and \textit{P. W. Jones} [Duke Math. J. 66, No. 2, 169-206 (1992; Zbl 0765.30005)].