Notes on the Duren–Leung conjecture

For the logarithmic coefficients γn of a univalent function f (z) = z+a2z2 +· · · ∈ S, the well-known de Branges’ theorem shows that Mn(f ) := 1 n n −1 m=1 m k=1 1 k − k|γk |2 0 (n = 2, 3, . . .). In this note, we first use properties ofMn(f ) to obtain some identities for γn, we then show that the Duren– Leung conjecture n k=1 |γk |2 n k=1 1/k2 (n 3) holds in the case when |a2| (4 − 75δ/26)1/2 = 1.76 . . . or when f is not Koebe function and n max{75δ/(26(1 −|γ1|2))−1, 3} is an integer, where δ is the Milin constant. Finally we give several remarks on a related question. © 2006 Elsevier Inc. All rights reserved