New inequalities for permanents and hafnians and some generalizations

Abstract: We show new upper bounds for permanents and hafnians, which are particularly useful for complex matrices. Multidimensional permanents and hyperhafnians are considered as well. The permanental bounds improve on a Hadamard type inequality of Carlen, Lieb and Loss (2006, Methods and Applications of Analysis 13, 1--17) and Cobos, K\"uhn and Peetre (2006, Integral Equations and Operator Theory 56, 57--70). Our proofs are based on a more general inequality, which can be applied to generalized Laplace type expansions of the matrix functions under consideration. As application, we show new bounds on the characteristic function of random diagonal sums. A numerical comparison shows the performance of some of our permanental bounds.