Inequalities for Indefinite Forms

Let nG2 be a fixed integer and Fp x.[ x1pyx2py???yxnp.1rp xgRp., where Rp denotes the set of all vectors xs x1, . . . , xn.for which xiG0 is 1, . . . , n., x1pGx2pq???qxnpif p)0 and xi)0 is1, . . . , n., x1p)x2pq???qxnp if p-0. Three inequalities are presented for Fp. The first is a comparison theorem. The second is a ‘‘H¨older-like’’ generalization of Acz´el’s inequality an extension of Popoviciu’s inequality., while the third is a generalization of Bellman’s inequality to all possible values of p. The proofs show that the above inequalities are consequences of some well-known inequalities for power means and power sums.