If A and B are self-adjoint matrices and $A leq B$,
what functions $f$ have the property that $f(A)$ must be less than
or equal to $f(B)$?
This question was answered by K. Lowner in 1934.
Recently J. Agler, N. Young and I have extended this result to functions
of more than one variable.
I will describe Lowner's results, and our extensions of them.