Proof of Lorch's conjecture on ultraspherical polynomials

We use a Volterra integral equation to derive lower bounds for the local maxima of |un(θ)| = (sin θ)λ|P(λ)n(cos θ)|, where P(λ)n (·) is the nth ultraspherical polynomial with parameter 0 < λ < 1. Moreover, inequalities for the critical points and inequalities between the extrema of un(θ) and un−1(θ) are obtained. The results are applied to show that, for every λ, the maxima of (n+λ)1−λ|un(θ)| form a strictly increasing sequence. This establishes a conjecture of Lorch [12, 13].