Jensen's inequality for operators and L�wner's theorem

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

“…As we mentioned in the first section, if f (t) ≥ 0 is continuous on [0, ∞), then f is operator monotone if and only if f is operator concave [4]. Now we slightly extend it to see that u −1 (s γ ) in Theorem 3.2 is operator concave.…”

“…It is known that each p n has n simple zeros and there is one zero of p n−1 between any two consecutive zeros of p n (see p. 61 of [3] (4). If the support of dµ is contained in (−∞, a], every zero of p n is in this interval.…”

“…The sum of operator monotone functions is also an operator monotone function. By the Löwner theorem [7] (see also [4], [6]), f is operator monotone if and only if f has an analytic extension f (z) to the open upper half plane Π + so that f (z) maps Π + into itself. Thus if f (t) ≥ 0 and g(t) ≥ 0 are operator monotone, so is f (t) µ g(t) λ for 0 ≤ µ, λ ≤ 1, µ + λ ≤ 1.…”

Operator Inequalities Associated with Jensen’s Inequality

“…The original proof by Löwner is related to the Cauchy Interpolation Problem, while the proof by Bendat and Sherman [8] is connected to the Hamburger Moment Problem. The proof by Korányi [40] and Sz-Nagy [56] makes use of the theory of selfadjoint operators on a Hilbert space, while the proof by the author and Pedersen [33] invokes the theorem of Krein-Milman.…”

“…The author and Pedersen [33] essentially proved that a mid-point matrix convex function f of order 2n defined on an interval 1 I with 0 ∈ I and f (0) ≤ 0 satisfies the inequality…”