On Turán inequality for ultraspherical polynomials

We show that the normalised ultraspherical polynomials, $G_n^{(\lambda )}(x)=C_n^{(\lambda )}(x)\mathrm{/}{C}_n^{(\lambda )}(1)$, satisfy the following stronger version of Turán inequality, $$\mathrm{\mid }x{\mathrm{\mid }}^{\theta }{(G_n^{(\lambda )}(x))}^2-G_{n-1}^{(\lambda )}(x)G_{n+1}^{(\lambda )}(x)\ge 0,\mathrm{\mid }x\mathrm{\mid }\le 1,$$ where $\theta =4\mathrm{/}(2-\lambda )$ if $-1\mathrm{/}2<\lambda \le 0$, and $\theta =2\mathrm{/}(1+2\lambda )$ if $\lambda \ge 0$. We also provide a similar generalisation of Turán inequalities for some symmetric orthogonal polynomials with a finite or infinite support defined by a three term recurrence.