Toward the best possible Rosenthal-type bound

An exact upper bound on the absolute moments of order p \geq 5 of sums of independent zero-mean random variables with fixed values of the sums of the absolute moments of orders 2 and p is obtained, in terms of two independent centered Poisson random variables. A somewhat more general result is in fact obtained. As a tool used in the proof, a calculus of variations of generalized moments of infinitely divisible distributions with respect to variations of the L\'evy characteristics is developed.