Optimal constants in the Rosenthal inequality for random variables with zero odd moments

We obtain estimates for the best constant in the Rosenthal inequality E|∑i=1nξi|2m⩽C(2m)max(∑i=1nEξi2m,(∑i=1nEξi2)m) for independent random variables ξ1,…,ξnξ1,…,ξn with ll zero first odd moments, l⩾1l⩾1. The estimates are sharp in the extremal cases l=1l=1 and l=ml=m, that is, in the cases of random variables with zero mean and random variables with mm zero first odd moments.