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In this paper we improve the estimates obtained in \\cite{AP2} for certain special classes of functions.\n  In particular, we improve estimates of Kato \\cite{Ka} and show that $$ \\big\\|\\,|S|-|T|\\,\\big\\|\\le C\\|S-T\\|\\log(2+\\log\\frac{\\|S\\|+\\|T\\|}{\\|S-T\\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\\df(S^*S)^{1/2}$. 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