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 thatMn(f):=1n∑m=1n−1∑k=1m(1k−k|γk|2)⩾0(n=2,3,…). In this note, we first use properties of Mn(f) to obtain some identities for γn, we then show that the Duren-Leung conjecture ∑k=1n|γk|2⩽∑k=1n1/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.