Uniform estimates of ultraspherical polynomials of large order

In this paper we prove the sharp inequality vertical bar P-n((s))(x)vertical bar P-n((s))(1)(vertical bar x vertical bar(n) + n-1/2s+1 (1 - vertical bar x vertical bar(n))), where P-n((s)) (x) is the classical ultraspherical polynomial of degree n and order s >= n 1+root 5/4. This inequality can be refined in [0, z(n)(s)] and [z(n)(s), 1], where z(n)(s) denotes the largest zero of P-n((s)) (x).