Journal article
Positivity and monotony properties of the de Branges functions
Publication Details
Authors:  Koepf, W.; Schmersau, D.

Journal:  Journal of Computational and Applied Mathematics 
Abstract
In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions tau(k)(n)(t) which follows from an identity about Jacobi polynomial sums that was published by Askey and Gasper in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Lambda(k)(n)(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', tau(k)(n)(t) = kLambda(k)(n)(t), and the positivity results in both proofs (n)(k)(t) less than or equal to 0 are essentially the same. By the relation tau(k)(n)(t) less than or equal to 0, the de Branges functions tau(k)(n)(t) are monotonic, and tau(k)(n)(t) greater than or equal to 0 follows. In this article, we reconsider the de Branges and Weinstein functions, find more relations connecting them with each other, and make the above positivity and monotony result more precise, e.g., by showing tau(k)(n)(t) greater than or equal to (nk+1)e(kt). (C) 2004 Elsevier B.V. All rights reserved.