S. Sadigova

The lower envelope of ultraspherical polynomials

Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $\alpha\ge0$, put $\xi_0=-1$ and let $\xi_n$ be the largest zero of $R_n^{(\alpha+1,\alpha)}$ for $n\ge1$. We prove that $$ R_k^{(\alpha,\alpha)}(x) =\min_{n\ge0}R_n^{(\alpha,\alpha)}(x), \quad \xi_{k-1}\le x\le\xi_k, $$ for every $k...