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} \Delta_n(x):=P_n^{(\alpha_n,\beta_n)}(x)P_n^{(\alpha_{n+1},\beta_{n+1})}(x)- P_{n-1}^{(\alpha_n,\beta_n)}(x)P_{n+1}^{(\alpha_{n+1},\beta_{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 Tu\'ran type inequalities, but with $\alpha_n$ and $\beta_n$ that depends linearly on $n$.