We study the Kraus matrices associated with the function f(x) = x root x - epsilon x(2) on (0, 1) for 1/2n+2 Superscript/Subscript Available</comment
We study the Kraus matrices associated with the function f(x) = x root x - epsilon x(2) on (0, 1) for 1/2n+2 < epsilon <= 1/2n+1 to show that f(x) is matrix convex of order n but not of order (n + 1) as an application of our preceding results on matrix monotone functions. (c) 2022 Elsevier Inc. All rights reserved. Superscript/Subscript Available
In this article, following [4,3], we consider matrix perspectives and show that the n-convexity of a function f on an interval I subset of (0, infinity) is equivalent to that of the function g(t) = tf (t-1) on I-1. (c...
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In this article, following [4,3], we consider matrix perspectives and show that the n-convexity of a function f on an interval I subset of (0, infinity) is equivalent to that of the function g(t) = tf (t-1) on I-1. (c) 2023 Elsevier Inc. All rights reserved.
Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi- Davis-Jensen inequality f (Phi(A,B)) <= Phi(f(A), f(B)) does not hold in general for a matrix con...
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Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi- Davis-Jensen inequality f (Phi(A,B)) <= Phi(f(A), f(B)) does not hold in general for a matrix convex function f and a positive multilinear mapping Phi. We prove this inequality under certain condition on f. Moreover, some Kantorovich and convexity type inequalities including positive multilinear mappings are presented. (C) 2015 Elsevier Inc. All rights reserved.
A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided p(tX + ( 1 - t) Y) less than or equal to tp( X) + ( 1 - t) p(Y) for all 0 less than or equal to t less than or equal to 1 a...
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A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided p(tX + ( 1 - t) Y) less than or equal to tp( X) + ( 1 - t) p(Y) for all 0 less than or equal to t less than or equal to 1 and for all tuples X = (X-1,..., X-g) and Y = (Y-1,..., Y-g) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension ( depending upon p). The main result of this paper is that every matrix convex polynomial has degree two or less. More generally, the polynomial p has degree at most two if convexity holds only for all matrices X and Y in an "open set." An analogous result for nonsymmetric variables is also obtained. matrix convexity is an important consideration in engineering system theory. This motivated our work, and our results suggest that matrix convexity in conjunction with a type of "system scalability" produces surprisingly heavy constraints.
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