An inequality for Jacobi polynomials of form $P_n^{({\alpha }_n,{\beta }_n)}(x)$

Abstract

We prove an inequality for Jacobi polynomials that \begin{align} Δ_n(x):=P_n^{(α_n,β_n)}(x)P_n^{(α_{n+1},β_{n+1})}(x)- P_{n-1}^{(α_n,β_n)}(x)P_{n+1}^{(α_{n+1},β_{n+1})}(x)\le 0,\ \forall x\ge 1, \end{align} where ${\alpha }_n=an$ and ${\beta }_n=bn$ for some $a,b\ge 0$. The above inequality has a similar taste as the Tuŕan type inequalities, but with ${\alpha }_n$ and ${\beta }_n$ that depends linearly on $n$.