Notes on the Duren-Leung conjecture

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:332

Article

Li, Jian-Lin

关键词: logarithmic coefficients;    de Branges' theorem;    Milin constant;   

DOI  :  10.1016/j.jmaa.2006.10.030

来源: Elsevier

【 摘 要 】

For the logarithmic coefficients gamma(n) of a univalent function f (z) = z + a(2)z(2) + ... is an element of S, the well-known de Branges' theorem shows that M-n(f):= 1/n Sigma(n-1)(m=1)Sigma(m)(k-1) (1/k - k vertical bar gamma k vertical bar(2)) >= 0 (n = 2,3,...). In this note, we first use properties of M-n(f) to obtain some identities for gamma(n), we then show that the Duren-Leung conjecture Sigma(n)(k=1)vertical bar gamma(k)vertical bar(2) <= Sigma(n)(k=1) 1/k(2) (n >= 3) holds in the case when vertical bar a(2)vertical bar <= (4 - 75 delta/26)(1/2) = 1.76... or when f is not Koebe function and n >= max [75 delta/(26(1 - vertical bar gamma(1)vertical bar(2))) - 1,3] is an integer, where delta is the Milin constant. Finally we give several remarks on a related question. (c) 2006 Elsevier Inc. All rights reserved.

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