Notes on the Duren-Leung conjecture
Assume that \(f(z)= z+ a_2z^2+\cdots\) is in the class \(S\) and \(\log{f(z)\over z}= 2\sum^\infty_{k=1}\gamma_k z^k\). Let \(\delta\) denote the Milin constant. The Duren-Leung conjecture asserts that \[ \sum^n_{k=1} |\gamma_k|\leq \sum^n_{k=1} k^{-2}\tag{\(*\)} \] and it remains open for \(n\geq 3\). The main objective of this note is to add some new observations concerning \((*)\). Main results of this paper are as follows: (1) if \(|a_2|\leq (4-75\delta/26)^{1/2}= 1.76\dots\) then \((*)\) holds for \(n\geq 3\), (2) if \(f(z)\) is not the Koebe function then there exists a constant \(N(f)\) such that \((*)\) holds for all \(n\geq N(f)\).
scientific article; zbMATH DE number 5156414
scientific article