Improving some operator inequalities for positive linear maps

Abstract

Let 0 < mI ≤ A ≤ m I ≤ M I ≤ B ≤ MI and p ≥ 1. Then for every positive unital linear map Φ, Φ 2p (A∇ t B) ≤ (K(h,2) 4 1 p −1 (1+Q(t)(log M m) 2)) 2p Φ 2p (A t B) and Φ 2p (A∇ t B) ≤ (K(h,2) 4 1 p −1 (1+Q(t)(log M m) 2)) 2p (Φ(A) t Φ(B)) 2p , where t ∈ [0, 1], h = M m , K(h, 2) = (h+1) 2 4h , Q(t) = t 2 2 (1−t t) 2t and Q(0) = Q(1) = 0. Moreover, we give an improvement for the operator version of Wielandt inequality. where µ ∈ [0, 1]. When µ = 1 2 , we write A∇B and A B for brevity, respectively, see [1] for more details. The Kantorovich constant is defined by K(t, 2) = (t+1) 2 4t for t > 0. A linear map Φ : B(H) → B(H) is called positive (strictly positive) if Φ(A) ≥ 0 (Φ(A) > 0) whenever A ≥ 0 (A > 0), and Φ is said to be unital if Φ(I) = I.

Further Improved Weighted Arithmetic-Geometric Operator Mean Inequalities

“…In this paper, we first present two weighted arithmetic-geometric operator mean inequalities, which refine and generalize inequalities (5) and (6), moreover, an example shows that inequalities (11) and (12) are sharper than inequalities (5) and (6), respectively. Finally, we generalize inequalities (11) and (12) to the power of p (p ≥ 2), which refine inequalities (3) and (4).…”

“…In what follows, when α = 1 2 , we present an example showing that inequalities (11) and (12) are sharper than inequalities (5) and (6), respectively.…”

“…Because of inequality (10), inequalities (11) and (12) are sharper than inequalities (5) and 6, respectively.…”

“…For further reading related to operator inequalities, the reader is referred to recent papers [5][6][7][8][9], and the references therein.…”

Further refinements of reversed AM–GM operator inequalities

“…In fact, we can get (1.10) from (1.12) and (1.11) from (1.13) when v = 1 2 , respectively. For more information about AM-GM operator inequalities, we refer the readers to [9,11,[13][14][15][16][17][18] and the references therein.…”