I want to show for $1\leq b <\infty$,
$$\left(\frac{\Gamma(\frac{a}2)\Gamma(\frac{b+1}2)}{\Gamma(\frac{1}2)\Gamma(\frac{a+b}2)}\right)^{1/b}\sim\frac{\sqrt{b}}{\sqrt{a+b}}$$ with absolute constants of equivalence.
I am trying to find some gamma function asymptotics but didn't have any idea. I also tried Gautschi’s Inequality but fail to prove it.
Any suggestions would be welcome and thanks to you!