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Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$ large enough. Let $\psi:[0,\infty)\to(0,1]$ be a smooth bijection with $\psi'<0$. Is there any $K>0$ such that $\psi(T_s)\le\psi(T_0+Ks^2)$ for $s$ large enough?

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Although this is not equivalent, it is related to the notion of operator monotone functions. OMF are very special functions, in particular they are analytic. $f(t)=t^2$ is not OM, but $\sqrt t$ is OM. For instance, in your case, you do have $$\sqrt{T_0+Cs^2}\le \sqrt{T_s}\le\sqrt{T_0+C's^2}.$$ –  Denis Serre Mar 7 '12 at 15:33

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