EXACT ROSENTHAL-TYPE BOUNDS

It is shown that, for any given p >= 5, A > 0 and B > 0, the exact upper bound on is an element of vertical bar Sigma X-i vertical bar(p) over all independent zero-mean random variables (r.v.'s) X-1, ..., X-n such that Sigma is an element of X-i(2) = B and Sigma is an element of vertical bar X-i vertical bar(p) = A equals c(p) is an element of vertical bar Pi(lambda) - lambda vertical bar(p), where (lambda, c) is an element of (0, infinity)(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, a more general result is obtained, as well as other related ones. As a tool used in the proof, a calculus of variations of moments of infinitely divisible distributions with respect to variations of the Levy characteristics is developed.