On majorization inequalities for normalized Jack and Macdonald polynomials
We study the inequality $P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)$ for all $x\in\mathbb{R}_{>0}^n$ whenever the partition $\lambda$ majorizes $\mu$; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every $n$ from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case $n=2$, the covers $(d)\succeq(d-1,1)$ for every $n$ when $t\leq q$, and, by exact certificates, the case $n=3$ with $|\lambda|\leq 8$. The HCIZ mechanism behind the Jack proof…
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