Exact Rosenthal-type bounds for $p\ge5$

It is shown that, for any given $p\ge5$, $A>0$, and $B>0$, the exact upper bound on $\operatorname{\mathsf{E}}|\sum X_i|^p$ over all independent zero-mean random variables (r.v.'s) $X_1,\dots,X_n$ such that $\sum \operatorname{\mathsf{E}} X_i^2=B$ and $\sum \operatorname{\mathsf{E}} |X_i|^p=A$ equals $c^p \operatorname{\mathsf{E}} |\Pi_{\lambda}-\lambda|^p$, where $(\lambda,c)\in(0,\infty)^2$ is the unique solution to the system of equations $c^p\lambda=A$ and $c^2\lambda=B$, and $\Pi_\lambda$ is a Poisson r.v.\ with mean $\lambda$. In fact, more general results are obtained. As a tool used in the proof, a calculus of variations of moments of infinitely divisible distributions with respect to variations of the L\'evy characteristics is developed.