A note on de Branges’ weights

Abstract

We discuss the de Branges weight systems and give a counterexample uncovering pitfalls. 1. PRELIMINARIES For a positive integer n and t > 0, a de Branges weight system is a system of nonincreasing functions o-~(t) satisfying the differential recurrence (1) am m+ = m+ m = 1,..., n, an+l0. The weight system defined by the initial conditions am (0) = n + 1 m plays a key part in de Branges' famous proof of the inequality E (n +1 -m)(mlcm2 4/m) < 0 m= 1 for the logarithmic coefficients Cm of normalized univalent functions, conjectured by Milin and implying the Bieberbach conjecture [1] (results, history and bibliography can be found in [2, 4]). In fact, de Branges' theorem contains the inequality (2) Xm(mlcml2 4/m) < 0 m= 1 for any xm that are the initial values am (0) of nonincreasing weights am. An explicit or asymptotic knowledge of numbers x1,... , , such that (2) holds potentially opens the door to other interesting estimates. Not all of these tuples arise as initial values of the nonincreasing solutions of (1), that is via de Branges' method, and their description is unknown. However, a complete description of all nonincreasing solutions of (1) is given in [3] in the framework of a more general setting. Because of certain implications, our attention has recently been drawn to [6] which claims to obtain (2), by de Branges' method, for any choice of the initial Received by the editors January 6, 2004. 2000 Mathematics Subject Classification. Primary 30C50, 30C55, 30C75.