Let $d$ and $M$ be two positive integers and let $0 < r < 1/2$. Is it possible to prove the following inequality:
$ (A):= \, \, \frac{M^{d-1} \pi^{d/2}d^{3/2}r^{d-1}}{2 \Gamma(1+d/2)} \leq \frac{\pi^{^{}-2}2^{-d}dr^{d-1}M^{d-1}}{1000} \, \, \, =:(B)$
(at least for large values of $d$)?
Actually the expressions $(A)$ and $(B)$ come out in the problem of comparing two quantities, say $D_{d,M}$ and $G_{d,M}$, with the aim of showing that $D_{d,M}<G_{d,M}$ for large values of $d$. In this view, the author first shows that, for every fixed $d$, $D_{d,M}<(A)$. Then he claims: "in order to conclude the proof we show that $G_{d,M}>(B)$". So I guess that the inequality $(A)<(B)$ holds true at least for large values of $d$. I tried to use Stirling's formula for the gamma function but I didn't get the result.