Positivity and monotony properties of the de Branges functions

Abstract

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions τkn(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 Λkn(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', τ˙kn(t)=-kΛkn(t), and the positivity results in both proofs τ˙kn(t)≤0 are essentially the same.By the relation τ˙kn(t)≤0, the de Branges functions τkn(t) are monotonic, and τkn(t)≥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 τkn(t)≥(n−k+1)e−kt.