  We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a
real function defined on the positive half-line, and $ A $ and $ B $ are
matrices such that $ A $ is positive definite and $ B $ is positive
semi-definite. If $ f $ is non-negative and operator monotone decreasing, then
such a trace function can be written as the Laplace transform of a positive
measure. The question is related to the Bessis-Moussa-Villani conjecture.
  Key words: Trace functions, BMV-conjecture.
