This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operatorconvexfunctions, and strongly operator convex functions. strongly operator convex functions were previous...
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This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operatorconvexfunctions, and strongly operator convex functions. strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We show also that if to is a point in the domain of a continuous function f, then f is operator monotone if and only if (f (t) - f (t(0)) / (t - t(0)) is stronglyoperatorconvex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operatorconvexity or strong operatorconvexity of phi o f when f is operatorconvex or stronglyoperatorconvex. (C) 2018 Elsevier Inc. All rights reserved.
The objective of this paper is to investigate operatorfunctions by making use of the operator harmonic mean '!'. For 0 = 0 in order that C 0 and g(A+B/2) 0, then f is operator monotone if and only if f (A! ...
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The objective of this paper is to investigate operatorfunctions by making use of the operator harmonic mean '!'. For 0 < A <= B, we construct a unique pair X, Y such that 0 < X <= Y, A = X!Y, B = X+Y/2. We next give a condition for operators A, B,C >= 0 in order that C <= A! B and show that g not equal 0 is stronglyoperatorconvex on J if and only if g(t) > 0 and g(A+B/2) <= g(A) ! g(B) for A, B with spectra in J. This inequality particularly holds for an operator decreasing function on the right half line. We also show that f(t) defined on (0, b) with 0 < b <= infinity is operator monotone if and only if f(0+) < infinity, f(A! B) <= 1/2(f(A) + f(B)). In particular, if f > 0, then f is operator monotone if and only if f (A! B) <= (A)! f (B). We lastly prove that if a stronglyoperatorconvex function g(t) > 0 on a finite interval (a, b) is operator decreasing, then g has an extension to (a, infinity) that is positive and operator decreasing.
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