Majorization and some operator monotone functions (Q848565)

Let \(h\) be a non-decreasing function on an interval \(I\) and \(k\) be an increasing function on an interval \(J\). Then \(h\) is said to be majorized by \(k\), denoted by \(h \preceq k\,\,(J)\), if \(J\subseteq I\) and the composite \(h\circ k^{-1}\) is operator monotone on \(k(J)\). The main result of the paper is that \(\frac{t-a}{f(t)-f(a)}\) is a nonnegative operator monotone function if \(f(t)\) is operator monotone on \((0, \infty)\) and \(0 <a<\infty\). The author then utilizes the notion of majorization to conclude that \(\frac{(t-a)(t-b)}{(t^r-a^r )(t^{1-r}-b^{1-r})}\) is operator monotone on \([0,\infty)\) for \(0 \leq a, b < \infty\) and for \(0 \leq r \leq 1\). The special case where \(a = b = 1\) gives rise to a theorem of \textit{D.\,Petz} and \textit{H.\,Hasegawa} [Lett.\ Math.\ Phys.\ 38, No.\,2, 221--225 (1996; Zbl 0855.58070)].