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Operator inequalities via geometric convexity. (English) Zbl 1435.47018

The authors give some norm and numerical radius inequalities for Hilbert space operators.
Firstly, they give upper bounds of \(f(w(B^{*}A))\) and \(f(w^{2}(A))\) for an increasing convex function \(f\), where \(A, B\) are bounded linear operators and \(w(A)\) means the numerical radius of an operator \(A\). They are generalizations of the results in [S. S. Dragomir, Sarajevo J. Math. 5 (18), 269–278 (2009; Zbl 1225.47008); F. Kittaneh, Stud. Math. 158, 11–17 (2003; Zbl 1113.15302); K. Shebrawi et al., J. Inequal. Appl. 2009, Article ID 492154, 11 p. (2009; Zbl 1179.47004)] and [M. El-Haddad et al., Stud. Math. 182, 133–140 (2007; Zbl 1130.47003)].
Secondly, the authors discuss the analogous argument for an increasing geometrically convex and concave function \(f\). Here, a function \(f\) is said to be geometrically convex if and only if \[ f(t_{1}^{1-\nu}t_{2}^{\nu})\leq f^{1-\nu}(t_{1})f^{\nu}(t_{2}) \] holds for \(\nu\in [0,1]\), \(t_{1}, t_{2}\geq 0\).
Next, the authors show that, for positive operators \(A,B\) and an operator \(X\), the function \(f(\nu)=|||A^{\nu}XB^{\nu}|||\) is geometrically convex for all unitarily invariant norms. Related results and generalizations are obtained.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A12 Numerical range, numerical radius
26A51 Convexity of real functions in one variable, generalizations
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