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.

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