Some operator inequalities via convexity

Consider a complex Hilbert space (H, 〈·, ·〉). Let B (H) denote the algebra of all bounded linear operators acting on (H, 〈·, ·〉) An operator A is said to be positive (denoted by A ≥ 0) if 〈Ax, x〉 ≥ 0 for all x ∈ H, and also an operator A is said to be strictly positive (denoted by A > 0) if A is positive and invertible. The Gelfand map f (t) 7→ f (A) is an isometrically ∗isomorphism between the C-algebra C (sp (A)) of continuous functions on the spectrum sp (A) of a self-adjoint operator A and the C-algebra generated by 1H and A. If f, g ∈ C (sp (A)), then f (t) ≥ g (t) (t ∈ sp (A)) implies that f (A) ≥ g (A). On the other hand, when A ∈ B(H) is such that RA > 0, then A is said to be accretive. When H is finite dimensional, we identify B(H) with the algebra Mn of all complex n × n matrices. Given a matrix monotone function f : (0,∞) → (0,∞) with f(1) = 1, and two accretive matrices A,B, there is a matrix mean associated with f , denoted by σf or σ, defined by [1]