Geodesic
An Operator Karamata Inequality
Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 4
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We present an operator version of the Karamata inequality. More precisely, we prove that if $A$ is a selfadjoint element of a unital $C^*$-algebra $\mathscr{A}$ ,ρ is a state on $\mathscr{A}$ , the functions f ;g are continuous on the spectrum σ(A) of A such that $0$ $m$ $1$ ≤$f(s)$ ≤ $M$ $1$ , $0$ $m$ $2$ ≤ $g(s)$ ≤ $M$ $2$ for all $s\in \sigma(A)$ and $K=\left(\sqrt{m_1m_2}+\sqrt{M_1M_2}\right)/\left(\sqrt{m_1M_2}+\sqrt{M_1m_2}\right)$ then

$K^{-2}\le \frac{\rho(f(A)g(A))}{\rho(f(A)) \rho(g(A))}\le K^2.$

We also give some applications.
Classification : Primary 47A63; Secondary 47B25, 15A60 
@article{BMMS_2014_37_4_a2,
     author = {M. S. Moslehian and M. Niezgoda and R. Raji\'c},
     title = {An {Operator} {Karamata} {Inequality}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2014},
     volume = {37},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2014_37_4_a2/}
}
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%J Bulletin of the Malaysian Mathematical Society
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M. S. Moslehian; M. Niezgoda; R. Rajić. An Operator Karamata Inequality. Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 4. http://geodesic.mathdoc.fr/item/BMMS_2014_37_4_a2/