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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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