An operator inequality and its consequences

Locally Strong Majorization and Commutativity in $$\varvec{C^{*}}$$ C ∗ -Algebras with Applications

“…In this context, we prove a Hardy-Littlewood-Pólya-Karamata type theorem for such kind of majorization (see Theorem 1). Next, we employ the property of commutativity of self-adjoint operators to establish certain generalizations of a Moslehian-Micić-Kian type inequality [11] (see Sect. 3).…”

Operator inequalities of Jensen type

“…The authors of [9] used inequality (1.3) to obtain some operator inequalities. In particular, they gave a generalization of the Petrović operator inequality as follows: Some other operator extensions of (1.4) can be found in [1,2,11].…”

“…If f is a convex function on an interval J containing m, M, thenf (λA + (1 − λ)D) ≤ λf (A) + (1 − λ)f (D) − δ f X ≤ λf (A) + (1 − λ)f (D) (3.3)for all (A, D) ∈ Ω and all λ ∈ [0, 1], where X If f is concave, then inequality (3.3) is reversed.Proof. Put n = 1 and let Φ be the identity map in Corollary 3.3 to getf A + D 2 ≤ f (A) + f (D) 2 − δ f X ≤ f (A) + f (D) 2for any (A, D) ∈ Ω, which implies (3.3) by the continuity of f .Regarding to obtain an operator version of (3.4), it is shown in[9] that if f : [0, ∞) → [0, ∞) is a convex function with f (0) ≤ 0, then f (A) + f (B) ≤ f (A + B)(3.4) for all strictly positive operators A, B for which A ≤ M ≤ A + B and B ≤ M ≤ A + B for some scalar M. We give a refined extension of this result as follows. Theorem 3.6.…”

“…Many authors tried to obtain some operator extensions of (1.4). In[10], it was shown thatf (A + B) ≤ f (A) + f (B) for all non-negative operator monotone functions f : [0, ∞) → [0, ∞) if and only if AB + BA is positive.Another operator extension of (1.4) was established in[9] Theorem D.[9, Corollary 2.9] If f : [0, ∞) → [0, ∞) is a convex function with f (0) ≤ 0, then f (A) + f (B) ≤ f (A + B) for all invertible positive operators A, B such that A ≤ MI ≤ A + B and B ≤ MI ≤ A + B for some scalar M ≥ 0.…”

Interpolating Jensen-type operator inequalities for log-convex and superquadratic functions

“…Recently, Moslehian et.al. [9], showed that if f : J → R is a continuous convex function and Φ is a unital positive linear map on B(H ), then [10].…”