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arXiv · 2606.25468

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

Geno Nikolov
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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.

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Geno Nikolov. 2026-06-24. An inequality for Jacobi polynomials: a complement to Finite Increment Theorem. https://arxiv.org/abs/2606.25468

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