Worked solution: De Branges's proof of the Bieberbach conjecture via Milin's conjecture (1984)
Step 5 of 8: Reducing monotonicity to a positivity statement about Jacobi polynomials
In plain words
Differentiation turns the monotonicity of into a special-function sign problem. Jacobi polynomials provide the right coordinates, but the conclusion concerns the combined expression, not each piece in isolation.
Detailed analysis
Differentiating the full functional along the Loewner flow produces a finite combination of Jacobi-polynomial terms. De Branges's calculation shows that the needed sign follows from the Askey--Gasper positivity theorem in the specific parameter range arising here; it does not assert that unrelated individual summands are each non-positive.
- Jacobi polynomial
- A classical family of orthogonal polynomials depending on two parameters, widely used in the theory of special functions, spherical harmonics, and hypergeometric series; here used to expand and simplify the derivative .