Inequalities about ratios of Gamma functions

Question

Let $\Gamma(z)=\int_0^{\infty}x^{z-1}e^{-x}dx$ be the Gamma function (for the purpose of this question, one may assume $z\in\mathbb R$). I'm interested in $\Gamma(z+1/2)/\Gamma(z)$ for $z\geq 2$. From Wikipedia I found that $\Gamma(z+\alpha)/\Gamma(z)\sim z^\alpha$ as $z\to\infty$ but I'm wondering whether there are finite-sample, explicit constant versions. In other words, can I find constants $0<c\leq C<\infty$ such that $$ cz^\alpha \leq \frac{\Gamma(z+\alpha)}{\Gamma(z)} \leq Cz^\alpha, \;\;\;\;\;\;\forall z\geq 2? $$

Answer 1

Would $$\frac{x}{\sqrt{x+1}}<\frac{\Gamma(x+1/2)}{\Gamma(x)}<\sqrt x \qquad (x>0)$$ be sufficient? Follows from https://dlmf.nist.gov/5.6#E4. This implies $$\sqrt{\frac{2}{3}}\sqrt x<\frac{\Gamma(x+1/2)}{\Gamma(x)}<\sqrt x$$ for all $x≥2$. You can find several results on the internet by searching for inequalities for the Wallis ratio.