# An inequality for Jacobi polynomials of form $P_n^{(\alpha_n,\beta_n)}(x)$

## TL;DR

This paper establishes a new inequality involving Jacobi polynomials with parameters linearly depending on the degree, extending classical inequalities and providing insights into their behavior for large degrees.

## Contribution

The paper proves a novel inequality for Jacobi polynomials with linearly varying parameters, generalizing Turán-type inequalities for this class of orthogonal polynomials.

## Key findings

- Inequality holds for all x ≥ 1 with linearly dependent parameters
- Extends classical Turán inequalities to parameter sequences depending on n
- Provides bounds relevant for asymptotic analysis of Jacobi polynomials

## 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 $n$.