Let $N,q$ be natural numbers. Find the best upper and lower bound (non-asymptotic) for $$ \frac{\Gamma(2q)}{\Gamma(q)}\frac{\Gamma(q+N/2)}{\Gamma(2(q+N/2))}=\frac{\prod_{k=1}^q(2q-(2k+1))}{\prod_{\ell=1}^{q+N/2}(2(q+N/2)-(2\ell+1))} $$
2026-03-26 15:17:15.1774538235
inequality for ratio of Gamma functions
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