Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants

Abstract

For the Jacobi family with parameters $(0,\beta )$, $\beta \ge -1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi--Radau functions. When two consecutive functions are parametrised by the same Prüfer phase, the function of higher degree attains that phase closer to $\theta =0$ and has strictly larger amplitude. In particular, the moduli of all corresponding relative extrema increase strictly with the degree. The lower bound $-1/3$ is sharp for this continuous statement: when $-1<\beta <-1/3$, the amplitude inequality is reversed at sufficiently small positive phases. This local reversal does not determine the optimal range for the discrete extremal inequalities. For $-1/3\le \beta \le 0$, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so $({\Lambda }_n^{(0,\beta )}{)}_{n\ge 0}$ is strictly increasing. At $\beta =0$ the weighted Radau functions reduce to $P_m^{(0,-1)}$. We thereby recover the theorem of Wong and Zhang and obtain, through an exact total-variation formula, a short proof of the Qu--Wong theorem on Legendre Lebesgue constants. The representation and the termwise comparison together realise the alternative-expression approach proposed by Qu and Wong, without asymptotic expansions, error bounds, or finite numerical verification. The proof is based on the equal-phase Prüfer architecture developed in the first author's earlier preprint arXiv:2608.01404; it applies that architecture to the problem posed by Qu and Wong.