The lower envelope of ultraspherical polynomials

Abstract

Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha \ge 0$, put ${\xi }_0=-1$ and let ${\xi }_n$ be the largest zero of $R_n^{(\alpha +1,\alpha )}$ for $n\ge 1$. We prove that \[ R_k^{(α,α)}(x) =\min_{n\ge0}R_n^{(α,α)}(x), \quad ξ_{k-1}\le x\leξ_k, \] for every $k\ge 1$. This determines the pointwise lower envelope of the normalised ultraspherical polynomials and answers Question~2 posed by F.~M. de Oliveira Filho in his 2009 thesis.