  In this note we settle two open problems in the theory of permanents by using
recent results from other areas of mathematics. Bapat conjectured that certain
quotients of permanents, which generalize symmetric function means, are
concave. We prove this conjecture by using concavity properties of hyperbolic
polynomials. Motivated by problems on random point processes, Shirai and
Takahashi raised the problem: Determine all real numbers $\alpha$ for which the
$\alpha$-permanent (or $\alpha$-determinant) is nonnegative for all positive
semidefinite matrices. We give a complete solution to this problem by using
recent results of Scott and Sokal on completely monotone functions. It turns
out that the conjectured answer to the problem is false.
