On an exact constant for the Rosenthal inequality

Let $\xi_1,\dots,\xi_n$ be independent random variables having symmetric distribution with finite $p$th moment, $2<p<\infty$. It is shown that the precise constant $C^*_p$ in Rosenthal's inequality
$$ \biggl\|\sum_{i=1}^n\xi_i\biggr\|\le C_p\max\biggl(\biggl\|\sum_{i=1}^n\xi_i\biggr\|_2,\biggl(\sum_{i=1}^n\|\xi_i\|_p^p\biggr)^{1/p}\biggr) $$
has the form
\begin{align*} C_p^*&=\biggl(1+\frac{2^{p/1}}{\pi^{1/2}}\Gamma\biggl(\frac{p+1}2\biggr)\biggr)^{1/p}, \qquad 2<p<4, C_p^*&=\|\xi_1-\xi_2\|_p, \qquad p\ge4, \end{align*}
where $\Gamma(\alpha)=\int_0^\infty x^{\alpha-1}e^{-x} dx$, and $\xi_1$, $\xi_2$ are independent Poisson random variables with parameter 0.5. It is proved also that
$$ \lim_{p\to\infty}C_p^*\frac{\ln p}p=\frac1e. $$
.