Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
Power series, Bieberbach conjecture and the de Branges and Weinstein functions
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
| Hi-index | 7.29 |
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 ≤ 0 are essentially the same.By the relation τkn(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.