Extending the Hölder type inequality of Blakley and Roy to non-symmetric non-square matrices (Q2841353)

Let \(D\in\mathbb{R}^{m\times n}\), \(u\in\mathbb{R}^n\), \(v\in\mathbb{R}^m\), \(\|u\|=\|v\|=1\), where \(\|\cdot\|\) is the norm corresponding to the standard inner product \(\langle\cdot,\cdot\rangle\). The author proves that (1)~\(\langle (D^TD)^\alpha u,u\rangle\geq\langle Du,v\rangle^{2\alpha}\) for all \(\alpha\in\mathbb{R}\) with \(\alpha>1\), and (2)~\(\langle(DD^T)^kDu,v\rangle\geq\langle Du,v\rangle^{2k+1}\) for all \(k\in\mathbb{N}\) if \(D,u,v\geq 0\) (\(\geq\) is understood entrywise). He also gives equality conditions. The well-known inequality (3)~\(\langle D^ku,u\rangle\geq\langle Du,u\rangle^k\), where \(m=n\), \(D,u\geq 0\) and \(D\) is symmetric, is thus generalized.NEWLINENEWLINE\smallskipNEWLINENEWLINEThe following is conjectured by \textit{A.~Sidorenko} [Graphs Comb. 9, No. 2, 201-204 (1993; Zbl 0777.05096)]. Let \(h\) be a bounded, nonnegative and measurable function with respect to the Lebesgue measure~\(\mu\) on \([0,1]^2\), and let \(G=(V,E)\) be a bipartite graph. Then NEWLINE\[NEWLINE \int\prod_{\{i,j\}\in E}h(x_i,y_j)d\mu^{|V|}\geq\Big(\int hd\mu^2\Big)^{|E|}, NEWLINE\]NEWLINE where \(|\cdot|\) denotes the cardinality. As an application of~(2), the author verifies this conjecture assuming that \(G\) is a path.NEWLINENEWLINEReviewer's comments: Can \(k\in\mathbb{N}\) in~(2) be generalized to \(\alpha\geq 1\)? The author credits~(3) to \textit{G.~R.~Blakley} and \textit{P.~Roy} [Proc. Am. Math. Soc.~16, 1244--1245 (1965; Zbl 0142.27003)]. Prior to them, \textit{H.~P.~Mulholland} and \textit{C.~A.~B.~Smith} [Am. Math. Mon. 66, 673--683 (1959); Erratum. Ibid. 67, 161 (1960; Zbl 0094.00903)] proved it (but their equality condition was not completely right).