  We prove new explicit asymptotic formulae between (geometric) eigenvalues in
max-algebra and classical distinguished eigenvalues of nonnegative matrices,
which are useful tools for transferring results between both settings. We
establish new inequalities for both types of eigenvalues of Hadamard products
and Hadamard weighted geometric means of nonnegative matrices. Moreover, a
version of the spectral mapping theorem for the distinguished spectrum is
pointed out.
