Some refinements of operator reverse AM-GM mean inequalities

摘要

In this paper, we prove the operator inequalities as follows: Let A, B be positive operators on a Hilbert space with 0  m ≤ A, B ≤ M and Mm≤2.314. Then for every positive unital linear map Φ, Φ2(A+B2)≤(M+m)24MmΦ2(A♯B) and Φ2(A+B2)≤(M+m)24Mm(Φ(A)♯Φ(B))2. Moreover, we prove Lin’s conjecture when Mm≤2.314.