An inequality of Turán type for Jacobi polynomials

Abstract:

For Jacobi polynomials $P_n^{(\alpha ,\beta )}(x),\alpha ,\beta > - 1$, let \[ {R_n}(x) = \frac {{P_n^{(\alpha ,\beta )}(x)}}{{P_n^{(\alpha ,\beta )}(1)}},\quad {\Delta _n}(x) = R_n^2(x) - {R_{n - 1}}(x){R_{n + 1}}(x).\] We prove that \[ {\Delta _n}(x) \geqq \frac {{(\beta - \alpha )(1 - x)}}{{2(n + \alpha + 1)(n + \beta )}}R_n^2(x),\quad - 1 \leqq x \leqq 1,n \geqq 1,\] with equality only for $x = \pm 1$. This shows that the Turán inequality ${\Delta _n}(\alpha ) \geqq 0, - 1 \leqq x \leqq 1$, holds if and only if $\beta \geqq \alpha > - 1$.