Operator monotone functions which are defined implicitly and operator inequalities

The function t(alpha) (0 < alpha < 1) is operator monotone on 0 less than or equal to t < infinity. This is known as the Lowner-Heinz inequality. However, not too many examples of concrete operator monotone functions are known so far. We will systematically seek operator monotone functions which are defined implicitly. This investigation is new, and our method seems to be powerful. We will actually find a family of operator monotone functions which includes t(alpha) (0 < alpha < 1). Moreover, by constructing one-parameter families of operator monotone functions, we will get many operator inequalities; especially, we will extend the Furuta inequality and the exponential inequality of Ando. (C) 2000 Academic Press.