### Abstract

Letn≥2 be a fixed integer andΦ_{p}(x):=(x_{1}^{p}-x_{2}^{p}-···-x_{n}^{p})^{1/p}(x∈R_{p}),whereR_{p}denotes the set of all vectorsx=(x_{1},...,x_{n}) for whichx_{i}≥0 (i=1,...,n),x_{1}^{p}≥x_{2}^{p}+···+x_{n}^{p}ifp>0 andx_{i}>0 (i=1,...,n),x_{1}^{p}>x_{2}^{p+}+···+x_{n}^{p}ifp<0. Three inequalities are presented for Φ_{p}. 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.

Original language | English |
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Pages (from-to) | 148-156 |

Number of pages | 9 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 205 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 1997 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*205*(1), 148-156. https://doi.org/10.1006/jmaa.1996.5188