  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 $n$.
