Archiv der Mathematik, ( ISI ), Volume (117), No (3), Year (2021-9) , Pages (347-359)

Title : ( The arithmetic-geometric mean inequality of indefinite type )

Authors: Mohammad Sal Moslehian , Takashi Sano , Kota Sugawara ,

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In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying I≥JA and sp(A)⊆[1,∞), the inequality I+A2≤JA−−√ holds, and the reverse does for A with I≥JA and sp(A)⊆[0,1] . We also prove that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality A+B2≥JA♯JB holds, where A♯JB:=J((JA)♯(JB)) . Several examples involving Pauli matrices are provided to illustrate the main results.


, J, selfadjoint matrix Indefinite inner product Arithmetic, geometric mean inequality
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author = {Sal Moslehian, Mohammad and Takashi Sano and Kota Sugawara},
title = {The arithmetic-geometric mean inequality of indefinite type},
journal = {Archiv der Mathematik},
year = {2021},
volume = {117},
number = {3},
month = {September},
issn = {0003-889X},
pages = {347--359},
numpages = {12},
keywords = {J-selfadjoint matrix Indefinite inner product Arithmetic-geometric mean inequality},


%0 Journal Article
%T The arithmetic-geometric mean inequality of indefinite type
%A Sal Moslehian, Mohammad
%A Takashi Sano
%A Kota Sugawara
%J Archiv der Mathematik
%@ 0003-889X
%D 2021