Trace functions as Laplace transforms

We study trace functions on the form $t\to \text{tr}f(A+tB)$ 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.