Tensorial and Hadamard product inequalities for functions of selfadjoint operators in Hilbert spaces in terms of Kantorovich ratio

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0 <γ ≤ f (t)/g (t)≤ Γ for t ∈ I and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then [f1−ν(A)g ν(A)] ⊗ [f ν(B)g 1−ν(B)] ≤ (1 − ν) f(A) ⊗ g (B) + νg(A) ⊗ f(B)                              ≤[(γ + Γ)2/4γΓ ]R [f1−ν (A) g ν(A)] ⊗ [f ν(B) g1−ν (B)]. The above inequalities also hold for the Hadamard product “ ◦ ” instead of tensorial product “ ⊗ ”.