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Worked solution: De Branges's proof of the Bieberbach conjecture via Milin's conjecture (1984)

Step 6 of 8: The Askey–Gasper inequality supplies the missing positivity
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In plain words

Askey and Gasper proved the Jacobi-polynomial positivity ∑j=0mPj(α,β)(x)Pj(β,α)(1)≥0\sum_{j=0}^{m} \frac{P_j^{(\alpha,\beta)}(x)}{P_j^{(\beta,\alpha)}(1)}\ge0 for β≥0\beta\ge0, α+β≥−2\alpha+\beta\ge-2, and −1≤x≤1-1\le x\le1. De Branges's coefficient calculation places the parameters in this theorem's admissible range, giving the sign Φn′(t)≤0\Phi_n'(t)\le0 needed in Step 5. The frequently displayed terminating 3F2{}_3F_2 form is an equivalent specialization, not an inequality valid for arbitrary unqualified parameters.

∑j=0mPj(α,β)(x)Pj(β,α)(1)≥0(β≥0, α+β≥−2, −1≤x≤1)(Askey–Gasper, 1976)\sum_{j=0}^{m} \frac{P_j^{(\alpha,\beta)}(x)}{P_j^{(\beta,\alpha)}(1)} \ge 0 \qquad (\beta\ge0,\ \alpha+\beta\ge-2,\ -1\le x\le1) \quad \text{(Askey--Gasper, 1976)}
Detailed analysis

Askey and Gasper proved the Jacobi-polynomial positivity ∑j=0mPj(α,β)(x)Pj(β,α)(1)≥0\sum_{j=0}^{m} \frac{P_j^{(\alpha,\beta)}(x)}{P_j^{(\beta,\alpha)}(1)}\ge0 for β≥0\beta\ge0, α+β≥−2\alpha+\beta\ge-2, and −1≤x≤1-1\le x\le1. De Branges's coefficient calculation places the parameters in this theorem's admissible range, giving the sign Φn′(t)≤0\Phi_n'(t)\le0 needed in Step 5. The frequently displayed terminating 3F2{}_3F_2 form is an equivalent specialization, not an inequality valid for arbitrary unqualified parameters.

Terms in this step
hypergeometric function 3F2{}_3F_2
A generalized hypergeometric series with three 'upper' and two 'lower' parameters, a classical special function that includes many combinatorial sums and orthogonal polynomial identities as special cases.
Askey–Gasper inequality
A 1976 positivity result of Richard Askey and George Gasper for a family of 3F2{}_3F_2 hypergeometric sums, originally proved for questions about positive definite functions, which turned out to be exactly the missing ingredient in de Branges's proof.
Knowledge used in this step
Special functions