Inequalities for the \(q\)-permanent. II
This paper is a sequel to the author's joint work with \textit{R. B. Bapat} [ibid. 197-198, 397-409 (1994; Zbl 0798.15004)]; see its review to glean undefined terms and conventions. Section 2: Th2: If \(A=BB^*\) with \(B\) a complex \(n\times r \) matrix, then for \(q\in [-1,1]\), there holds \( \Pi_q(A) \geq \sum_{\mathbf k}{1\over k_1!! \cdots k_r!!} \Pi_q (B[1,2,\ldots,n| {\mathbf k}]) \Pi_q (B^* [1,2,\ldots,n| {\mathbf k}])\) with inequality if \(q=\pm 1\). This generalizes a result of \textit{J. W. Bunce} [Linear Multilinear Algebra 30, No. 1/2, 5-11, (1991; Zbl 0742.15008)] and implies the Gram-type inequality of Bapat and Lal [loc. cit.]. Section 3: Denote by \(G_{k,n}\) the set of all sequences \(\alpha=(\alpha_i)_1^k\) of integers satisfying \(1\leq \alpha_1\leq \dots\leq \alpha_k \leq n\). Let \(\mu_q(\alpha)\) denote the product of the generalized factorials of the multiplicities of the distinct integers in \(\alpha\). Given an \(n\times n\) complex \(A\) define the \({n+k-1 \choose k}\)-square matrix \( P_{k,q}(A)\), called here k-th induced matrix, of entries \({\text{per}_q (A[\alpha| \beta]) \over \sqrt{\mu_q(\alpha) \mu_q(\beta)}}\), arranged lexicographically. The map \(P_{k,q}(\bullet)\) has obvious elementary properties. Th4: If \(A,B\geq 0\), then \(P_{k,q}(A)\geq 0\); for \(\ell\in \{1,k\}\), \(P_{k,q}((A+B)^{1/\ell})\geq P_{k,q}(A^{1/\ell})+ P_{k,q}(B^{1/\ell})\). The proof makes use of the concavity of \(A\mapsto \otimes^k A^{1/k}\); see \textit{T. Ando} [Linear Algebra Appl. 26, 203-241 (1979; Zbl 0495.15018)]. Th4 implies generalizations of results of \textit{T. Ando} [Hokkaido Math. J. 10, No. 1, 18-36 (1981; Zbl 0484.15006)] to the \(q-\)permanent. Section 4: The Cauchy-type inequalities of Bapat and Lal[loc.cit] is generalized, as are certain results found in connection with work prior to Falikman's 1981 proof of the van der Waerden permanent conjecture; e.g. Th13: If \(A\) is a positive semidefinite \(n\times n\) matrix with row sums \(r_1,\ldots, r_n\), and \(s(A)=\sum_{ij} a_{ij}\neq 0\), then \(\text{per}_q (A)\geq { n!! \over s(A)^n}\prod_{i=1}^n | r_i| ^2\). See e.g. \textit{M. Marcus} and \textit{H. Minc} [Trans. Am. Math. Soc. 116, 316-329; (1965; Zbl 0178.35902)] and other papers of these authors in that period.
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