ON REVERSE AM-GM INEQUALITIES FOR n OPERATORS

Abstract

In this paper, we generalize some operator inequalities due to Fu and He [Linear Multilinear Algebra, 63 (2015), 571-577] as follows: Let A_i (i = 1,...,n) be positive operators on a Hilbert space with 0 < m ≤ A_i ≤ M .  Then for every positive unital linear map Φ,

     Φ2p \(\left(\frac{A_1 + ... + A_n}{n}\right)\) ≤ \(\left[\frac{(M + m)^{2_p}}{4M^{p}m^{p}}\right]^{2}\) Φ^2p[G(A₁ , ... , A_n)], 1 ≤ p < \(\infty\),

and

     Φ2p \(\left(\frac{A_1 + ... + A_n}{n}\right)\) ≤ \(\left[\frac{(M + m)^{2_p}}{4M^{p}m^{p}}\right]^{2}\) G^2p[Φ(A₁) , ... ,  Φ(A_n)], 1 ≤ p < \(\infty\),

where G(A₁ , ... , A_n) is Ando-Li-Mathias geometric mean.