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

Abstract

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