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- Title
ON A CONJECTURE RELATED TO THE GEOMETRIC MEAN AND NORM INEQUALITIES.
- Authors
FREEWAN, SHAIMA’A; HAYAJNEH, MOSTAFA
- Abstract
A conjecture of Dinh, Ahsani, and Tam, was recently settled in [7]. In this note, we give a refinement to that result, namely if Ai and Bi are positive definite matrices and Z = [Zij] is the block matrix such that Zij = Bi1/2 (m/Σ/k=1 Ak) B1/2j for all i,j = 1, ...,m, then |||m/Σ/i=1 ( A²i # B² i )r ||| ≤ ||| Zr ||| ≤ ||| ((m/Σ/i=1 Ai)rp/2 (m/Σ/i=1 B²)rp (m/Σ/i=1 Ai)rp/2 )1/p |||, for all unitarily invariant norms, for all p > 0 and r ≥ 1 such that rp ≥ 1. Our approach provides us with an alternative proof without using the method of majorization that was used in [7]. As a byproduct, we get a refinement to a result of Audenaert in 2015.
- Publication
Mathematical Inequalities & Applications, 2024, Vol 27, Issue 1, p193
- ISSN
1331-4343
- Publication type
Academic Journal
- DOI
10.7153/mia-2024-27-15