Some Slater Type Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Some inequalities of the Slater type for convex functions of selfad-joint operators in Hilbert spaces H under suitable assumptions for the involved operators are given. Amongst others, it is shown that if A is a positive definite operator with Sp (A) ⊂ [m, M ] and f is convex and has a continuous derivative on [m, M ] , then for any x ∈ H with x = 1 the following inequality holds: 0 ≤ f Af ′ (A) x, x f ′ (A) x, x − f (A) x, x ≤ 1 4 · M f ′ (M) mf ′ (m) (M − m) f ′ (M) − f ′ (m) .