Higher Order Turán Inequalities

The celebrated Turán inequalities $P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0, \ x \in [-1,1], \ n \geq 1$, where $P_{n}(x)$ denotes the Legendre polynomial of degree $n$, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities $\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0, \ n \geq 1$, which hold for the Maclaurin coefficients of the real entire function $\psi$ in the Laguerre-Pólya class, $\psi(x) = \sum _{n=0}^{\infty} \gamma _{n} x^{n}/n!$.