An inequality involving the ℓ1, ℓ2, and ℓ NORMS

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For x [0, 1]n with 1 = 1 and y ∈ [1, ∞)n, we prove that $$\frac{\Vert\textit{\ textbf{xy}}\Vert-{\infty}}{\Vert\textit{\textbf{xy}}\Vert-{2}} \le \Vert\textit{\textbf{x}}\Vert-{\infty} \Vert\textit{\textbf{y}}\Vert-{1} \frac{\Vert\textit{\textbf{y}}\Vert-{\infty}}{\Vert\textit{\textbf{y}}\Vert-{2}} $$, where xy n is the vector with components x iyi. This bound does not seem to easily follow from known inequalities, and the proof technique may be of independent interest.

Original languageEnglish
Article number1350026
Pages (from-to)1-5
JournalAnalysis and Applications
Issue number6
StatePublished - 1 Nov 2013


  • Vector
  • inequalities
  • norm

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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