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Singular value inequalities involving convex and concave functions of positive semidefinite matrices. (English) Zbl 1446.15004
Summary: Let $$A$$ and $$B$$ be $$n\times n$$ positive semidefinite matrices, and let $$\alpha,\beta\in (0,1)$$ such that $$\alpha+\beta=1$$. Among other inequalities, it is shown that
(a)
If $$f$$ is a non-negative concave function on $$[0,\infty)$$, then $s_j(\alpha f(A)+\beta f(B))\leq s_j(f(\sqrt{2}\left|\alpha A+i\beta B\right|))$ for $$j=1,\dots,n$$.
(b)
If $$f$$ is a non-negative strictly increasing convex function on $$[0,\infty)$$ with $$f(0)=0$$, then $s_j(f\left(\alpha A+\beta B\right))\leq\sqrt{2}s_j(\alpha f\left(A\right) +i\beta f\left(B\right))$ for $$j=1,\dots ,n$$. Here $$s_j\left(X\right)$$ denotes the largest $$j$$th singular value of the matrix $$X$$.
##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 15A42 Inequalities involving eigenvalues and eigenvectors 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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