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Let A and B be n×n positive semidefinite matrices, and let ∥·∥2 be the Hilbert-Schmidt norm. Bhatia and then Hayajneh and Kittaneh, usingdi¤erent techniques, proved that
∥AvB1−v + BvA1−v∥2 ≤ ∥A + B∥2
for v ∈ [1/4, 3/4], which gives an affirmative answer to an open problem posed by Bourin for the special case of the Hilbert-Schmidt norm.
In this paper, we prove that
∥A v/2B1−v/2 + Bv/2 A1−v/2∥22 ≤1/2(∥A + B∥22 + ∥Bv/2 A1−vBv/2 + Av/2B1−vAv/2∥22)
for v ∈ [0, 1]. As a consequence of the this inequality, we give a new proof of the above inequality. We also prove that if r ≥ 1, then
|||AvB1-v + BvA1−v||| ≤ |||(A1/r + B1/r)r|||
for 1/2r ≤ v ≤ 2r−1/2r , where ||| · ||| denotes any unitarily invariant norm.
2020 Mathematics Subject Classification. Primary 15A60; Secondary 15B57; 47A30; 47B15.