A proof of Gautschi's conjecture on subrange Jacobi polynomials
Let $π_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^α(1+x)^β$, where $-1<α<β$. Gautschi conjectured that $$ \left[ \frac{π_n(-c)}{π_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{β-α} <1. $$ By his variation formula, this inequality is sufficient for every positive zero of $π_n$ to move to the right as $c$ increases. For $0<c<1$, a first-crossing argument proves the conjecture throughout $0<α<β$. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles $β\geq0$. In the negative wedge, writing $α=-r-λ$ and $β=-r+λ$, an ensemble Ward bound yields the region $c^2\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\geq1$, $-1<α<β$, and $0<c\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.