# inequality for ratio of Gamma functions

Let $$N,q$$ be natural numbers. Find the best upper and lower bound (non-asymptotic) for $$\frac{\Gamma(2q)}{\Gamma(q)}\frac{\Gamma(q+N/2)}{\Gamma(2(q+N/2))}=\frac{\prod_{k=1}^q(2q-(2k+1))}{\prod_{\ell=1}^{q+N/2}(2(q+N/2)-(2\ell+1))}$$