I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}$$ where $$\Delta_a=\{(x_1,...,x_n):0<x_1<a,0<x_2<1,...,0<x_n<1\}$$ and $1>a>b>0$, $\alpha>\frac{n+1}{2}$.

My question is, how can we upper bound $f_n(a,b)$? Is there a bound like some powers of $Ca/b$ for some constant $C>0$?

I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}$$ where $$\Delta_a=\{(x_1,...,x_n):0<x_1<a,0<x_2<a,...,0<x_n<a\}$$ and $1>a>b>0$, $\alpha>\frac{n+1}{2}$.

My question is, how can we upper bound $f_n(a,b)$? Is there a bound like some powers of $Ca/b$ for some constant $C>0$?