I asked the following question on stackexchange some time ago. Let $X,Y\in \mathbb{R}^d$ with $d\ge 1$. Also $X\sim N(0, \Sigma)$ and $Y$ (general r.v.) has zero mean and the same variance $\Sigma$. I'm looking for the following bound:

$$ \mathbb{E}\|X\|_{\infty}^r\le C_{d,r}\mathbb{E}\|Y\|_{\infty}^r $$

for some constant $C_{d,r}$ depending on the dimension of vectors $X$ and $Y$ ($d$) and $r\ge 2$. It's easy to see that we have

$$ \mathbb{E}\|X\|_{\infty}^r \le C_r d^{r/2}\mathbb{E}\|Y\|_{\infty}^r, $$

where $C_r$ is a constant depending on $r$. The latter follows from the fact that $\|x\|_{\infty}\le \|x\|_2\le \sqrt{d}\|x\|_{\infty}$ for $x\in \mathbb{R}^d$ and a known inequality regarding $\|\cdot\|_2$, i.e. $$ \mathbb{E}\|X\|_2^r\le C_r'\left(\mathbb{E}\|X\|_2^2\right)^{r/2}. $$

However, can we have $C_{d,r}$ of order $C_r \ln(d)$ or smaller?