A new majorization between functions, polynomials, and operator inequalities (Q2368773)

Consider for \(-\infty<a<b\leq\infty\) the set \(P_+[a,b)\) of non-negative operator monotone functions defined in the interval \([a, b)\). Denote with \( P_+^{-1}[a,b)\) the set set of increasing functions defined in the interval \([a, b)\) with range \([0,\infty)\) such that the inverse functions are in \(P_+[0,\infty)\). The main result of the paper under review is a proof of the inclusions \[ P_+[a,b)\cdot P_+^{-1}[a,b)\subset P_+^{-1}[a,b)\quad\text{and}\quad P_+^{-1}[a,b)\cdot P_+^{-1}[a,b)\subset P_+^{-1}[a,b) \] which are applied to polynomials of certain types. The resulting theory is then applied to various operator inequalities, and it is noted that Furuta's inequality appears as a special case.