A real-valued function, F, on an interval (a,b) is called matrix monotone
if F(A) < F(B)
whenever A and B are finite matrices of the same order with eigenvalues in (a,b) and A < B.
In 1934, Loewner proved the remarkable theorem that F is matrix monotone if and only if F
is real analytic with continuations to the upper and lower half planes so that Im F > 0 in the
upper half plane.
This deep theorem has evoked enormous interest over the years and a
number of alternate proofs.
There is a lovely 1954 proof that seems to have been "lost" in that the proof is not mentioned in
various books and review article presentations of the subject, and I have found no references to
the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof
and a variant of that proof
which I've found, which avoids the need for estimates, and proves a stronger theorem.