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

Abstract

Let $P=P_n^{(\alpha,\beta)}$ be the $n$-th degree Jacobi polynomial, which is orthogonal in $[-1,1]$ with respect to the weight function $(1-x)^{\alpha}(1+x)^{\beta}$, $\alpha,\beta>-1$. For parameters $(\alpha,\beta)$ satisfying either $\alpha\geq\beta\geq 1/2$ or $\alpha\geq 1/2$, $\beta=-1/2$, 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.