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Sharp inequalities for the permanental dominance conjecture. (English) Zbl 1215.15008
The permanental dominance conjecture asserts that the permanent has the largest value of any normalized generalized matrix function when evaluated at any positive semi-definite Hermitian matrix. The conjecture is known to hold for matrices of order 3, which is the case considered here. The author aims to show a stronger inequality than the permanental dominance conjecture. Specifically, if $$d(\cdot)$$ is a normalized generalized matrix function corresponding to a non-trivial irreducible character of the alternating group $$A_3$$ then
$d(H) \leq 2^{-1/3}\,\mathrm{per}\,H+(1-2^{-1/3})\det H$
for any $$3\times3$$ positive semi-definite Hermitian matrix $$H$$.
##### MSC:
 15A15 Determinants, permanents, traces, other special matrix functions 15A45 Miscellaneous inequalities involving matrices