Abstract: | Exact comparisons are made relating E|Y0|p, E|Yn−1|p, and E(maxj≤n−1 |Yj|p), valid for all martingales Y0,…,Yn−1, for each p ≥ 1. Specifically, for p > 1, the set of ordered triples {(x, y, z) : X = E|Y0|p, Y = E |Yn−1|p, and Z = E(maxj≤n−1 |Yj|p) for some martingale Y0,…,Yn−1} is precisely the set {(x, y, z) : 0≤x≤y≤z≤Ψn,p(x, y)}, where Ψn,p(x, y) = xψn,p(y/x) if x > 0, and = an−1,py if x = 0; here ψn,p is a specific recursively defined function. The result yields families of sharp inequalities, such as E(maxj≤n−1 |Yj|p) + ψn,p*(a) E |Y0|p ≤ aE |Yn−1|p, valid for all martingales Y0,…,Yn−1, where ψn,p* is the concave conjugate function of ψn,p. Both the finite sequence and infinite sequence cases are developed. Proofs utilize moment theory, induction, conjugate function theory, and functional equation analysis. |