Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation (Q935371)

The author gives some examples of operator monotone functions by an elementary method making use of the Löwner-Heinz inequality: \(A\geq B\geq 0\) ensures that \(A^{\alpha}\geq B^{\alpha} \) for \(\alpha \in [0,1]\) without appealing to Löwner's integral representation. The author uses the following nice variations of the Löwner-Heinz inequality: Let \(\alpha_{j}, \beta_{j},\gamma_{j},\dots\in [0,1]\) for \(j=1,2,\dots,n\). Then both (LH--1) and (LH--2), defined below, are operator monotone functions: \[ \left(\frac{1}{t^{\alpha_{1}} +\dots+ t^{\alpha_{n}}} +\frac{1}{t^{\beta_{1}} +\dots+ t^{\beta_{n}}}+ \frac{1}{t^{\gamma_{1}} +\dots+ t^{\gamma_{n}}}+\dots\right)^{-1} \tag{LH--1} \] and \[ (1+t^{-1})^{-\alpha_{1}}+(1+t^{-1})^{-\alpha_{2}}+\dots+(1+t^{-1})^{-\alpha_{n}}. \tag{LH--2} \] In particular, \((t^{-\alpha_{1}}+t^{-\alpha_{2}}+\dots+t^{-\alpha_{n}})^{-1}\) is operator monotone by (LH--1). As examples, the author states the following operator monotone functions: \(f(t)=\frac{1}{(1+t) \log (1+\frac{1}{t})} \), \( f(t)=\frac{t-1-\log t}{\log^2 t}\), and \(f(t)=\frac{t(t+2)}{(t+1)^2} \log (t+2)\), etc. Finally, the author gives an elementary proof of the known result that \(f_{p}(t)=\frac{p-1}{p} (\frac{t^p-1}{t^{p-1} -1})\) is an operator monotone function for \(-1\leq p\leq 2\).