Letn≥2 be a fixed integer andΦp(x):=(x1p-x2p-···-xnp)1/p(x∈Rp),whereRpdenotes the set of all vectorsx=(x1,...,xn) for whichxi≥0 (i=1,...,n),x1p≥x2p+···+xnpifp>0 andxi>0 (i=1,...,n),x1p>x2p++···+xnpifp<0. Three inequalities are presented for Φp. The first is a comparison theorem. The second is a "Hölder-like" generalization of Aczél's inequality (an extension of Popoviciu's inequality), while the third is a generalization of Bellman's inequality to all possible values ofp. The proofs show that the above inequalities are consequences of some well-known inequalities for power means and power sums.
ASJC Scopus subject areas
- Applied Mathematics