An inequality for Jacobi polynomials: a complement to Finite Increment Theorem

Abstract

Let $P=P_n^{(\hat{I}\pm ,{\hat{I}}^2)}$ be the $n$-th degree Jacobi polynomial, which is orthogonal in $[-1,1]$ with respect to the weight function $(1-x{)}^{\hat{I}}\pm (1+x{)}^2$, $\hat{I}\pm ,{\hat{I}}^2>-1$. For parameters $(\hat{I}\pm ,{\hat{I}}^2)$ satisfying either $\hat{I}\pm \ge {\hat{I}}^2\ge 1/2$ or $\hat{I}\pm \ge 1/2$, ${\hat{I}}^2=-1/2$, we prove the inequality $$P(1)-P(x)\ge P^{\prime }(x)(1-x),x\in [0,1],$$ which may be viewed as a complement to Finite Increment Theorem for Jacobi polynomials.

6 pages