matrix monotone

A real function $f$ on a real interval $I$ is said to be matrix monotone of $n$ , if

A\\leq_{L}B\\Rightarrow f(A)\\leq f(B)

(1)

for all Hermitian $n\\times n$ matrices $A, B$ with spectra contained in $I$ . Here $\\leq_{L}$ denotes the Loewner order, and the notation $f(A)$ is explained in the entry functional calculus for Hermitian matrices.

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