A Bernstein-type inequality for the Jacobi polynomial

Abstract:

Let $P_n^{(\alpha ,\beta )}(x)$ be the Jacobi polynomial of degree n. For $- \frac {1}{2} \leq \alpha ,\beta \leq \frac {1}{2}$, and $0 \leq \theta \leq \pi$, it is proved that \[ {(\sin \frac {\theta }{2})^{\alpha + \frac {1}{2}}}{(\cos \frac {\theta }{2})^{\beta + \frac {1}{2}}}|P_n^{(\alpha ,\beta )}(\cos \theta )| \leq \frac {{\Gamma (q + 1)}}{{\Gamma (\frac {1}{2})}}\left ( {\begin {array}{*{20}{c}} {n + q} \\ n \\ \end {array} } \right ){N^{ - q - \frac {1}{2}}},\] where $q = \max (\alpha ,\beta )$ and $N = n + \frac {1}{2}(\alpha + \beta + 1)$. When $\alpha = \beta = 0$, this reduces to a sharpened form of the well-known Bernstein inequality for the Legendre polynomial.