A mixed hölder and minkowski inequality

Alfredo N. Iusem, Carlos A. Isnard, Dan Butnariu

Research output: Contribution to journalArticlepeer-review


Hölder's inequality states that ∥x∥p ∥y∥q - 〈z, y〉 ≥ 0 for any (x, y) ∈ £p(Ω) × £q(Ω) with 1/p + 1/g = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q-2: ∥x∥p ∥y∥q, - 〈x, y〉 ≥ (1/P)[(∥x∥p + ∥z∥p)p - ∥x + z∥pp] ≥ 0 if P ∈ (1,2], 0 ≤ ∥x∥p ∥y∥q - 〈x, y〉 ≤ (1/p) [(∥x∥p + ∥z∥p)p - ∥x + z∥pp] if p ≥ 2. A similar result holds for complex valued functions with Re(〈x, y〉) substituting for 〈x,y〉. We obtain these inequalities from some stronger (though slightly more involved) ones.

Original languageEnglish
Pages (from-to)2405-2415
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number8
StatePublished - 1999
Externally publishedYes


  • Banach spaces
  • Holder's inequality
  • Minkowski's inequality

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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