Let \(X_ i\), \(i\geq 1\), be independent random variables with zero mean, \(S_ n=X_ 1+...+X_ n\), \(A_{t,n}=E(| X_ 1|^ t+...+| X_ n|^ t)\), \(B_ n=A^{1/2}_{2,n}\). An exact (but complicated) upper bound for \(E| S_ n|^ t\) is given in terms of \(A_{t,n}\) and \(B_ n\). The optimality of other earlier proved (and more simple) upper bounds for \(E| S_ n|^ t\) is investigated.