A problem for the coefficients of p-fold symmetric univalent functions (Q1077575)

Associated with each univalent function \(f\in S\) are its logarithmic coefficients \(\gamma_ n\) defined by log\(\frac{f(z)}{z}=2\sum^{\infty}_{n=1}\gamma_ nz^ n\); and the coefficients \(c_ n^{(p)}\) of its pth-root transform \((p=1,2,...)\), defined by \[ \{f(z^ p)\}^{1/p}=z+\sum^{\infty}_{k=1}c^{(p)}_{pk+1}z^{pk+1}. \] If f is the Koebe function, then \(\gamma_ n=1/n\) and \(c_ n^{(p)}=O(n^{2/p-1})\). This suggests the two conjectures that \(\gamma_ n=O(1/n)\) and \(c_ n^{(p)}=O(n^{2/p-1})\) for every \(f\in S\). The latter is known as the Szegö conjecture. It was proved by Littlewood for \(p=1\), by Littlewood and Paley for \(p=2\), by V. Levin for \(p=3\), and quite recently by A. Baernstein for \(p=4\). However, Littlewood disproved it for sufficiently large p, and Pommerenke disproved it for all \(p\geq 12\). (For \(5\leq p\leq 11\) it remains unsettled.) Pommerenke's construction also disproves the conjecture \(\gamma_ n=O(1/n)\); in fact, it produces a function \(f\in S\) with \(\gamma_ n\neq O(n^{-0.83})\). [See the reviewer's book ''Univalent functions'' (1983; Zbl 0514.30001), {\S} 8.1 for further details.] The author now exhibits a close connection between the two conjctures. His main result is that if a function \(f\in S\) satisfies \(\gamma_ n=O(1/n)\), then \(c_ n^{(p)}=O(n^{2/p-1})\) for \(p=1,2,... \). He also gives a partial converse: If \(| c_ n^{(p)}| \leq A_ pn^{2/p-1}\) for infinitely many p, and if \(\limsup_{p\to \infty}pA_ p=2a<\infty\) through this sequence of integers p, then \(\gamma_ n\leq a/n\).