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On some new converses of convex inequalities in Hilbert space. (English) Zbl 1325.47036
Assume that $$f:[m,M]\to\mathbb R$$ is a convex function, $$y_1,\dots,y_n\in[m,M]$$ are real numbers and $$p_1,\dots,p_n\in\mathbb R^{\geq 0}$$ such that $$P_n=\sum_{i=1}^np_i>0$$. The Lah-Ribarič inequality states that $\frac{1}{P_n}\sum_{i=1}^n p_if(y_i)\leq \frac{M-\overline{y}}{M-m}f(m)+\frac{\overline{y}-m}{M-m}f(M), \tag{1}$ where $$\overline{y}=\frac{1}{P_n}\sum_{i=1}^np_iy_i$$.
An operator version of Jensen’s inequality due to Mond and Pečarić says that, if $$A\in \mathbb B(\mathcal H)$$ is a selfadjoint operator with $$sp(A)\subseteq[m,M]$$ for some scalars $$m<M$$, and $$f:[m,M]\to\mathbb R$$ is a convex function, then $f\big(\langle Ax,x \rangle\big) \leq \langle f(A)x,x\rangle \tag{2}$ for each unit vector $$x\in \mathcal H$$, where $$\mathcal H$$ is a Hilbert space and $$\mathbb B(\mathcal H)$$ is the $$C^*$$-algebra of all bounded linear operators on $$\mathcal H$$.
Under the above assumptions, the authors prove the following inequalities for the difference between the right and the hand left side of ({2}): \begin{aligned} 0 & \leq \langle f(A)x,x\rangle - f\big(\langle Ax,x \rangle\big)\\ & \leq (M-\langle Ax,x\rangle)(\langle Ax,x\rangle -m)\frac{f'_-(M)-f'_+(m)}{M-m}\\ & \leq \frac{1}{4}(M-m)\left(f'_-(M)-f'_+(m)\right),\end{aligned}\tag{3} where $$f$$ is a continuous convex function on an interval of real numbers $$I$$ such that $$[m,M]$$ is a subset of the interior of $$I$$.
In the paper under review, the authors give refinements and improvements of converses of Jensen’s inequality ({2}) and also a refinement of ({3}) and an improvement of the scalar Lah-Ribarič inequality ({1}) for selfadjoint operators.
##### MSC:
 47A63 Linear operator inequalities 47A64 Operator means involving linear operators, shorted linear operators, etc.
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##### References:
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