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

Suppose $m$, $n$, and $k$ are positive integers, and let $\langle \cdot ,\cdot \rangle$ denote standard inner product on the spaces $\mathbb{R}^p$, $p\!>\!0$. We show that if $D$ is an $m\!\times \!n$ non-negative real matrix, and $u$ and $v$ are non-negative unit vectors in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, then

$\langle (D\!D^t)^k D u ,v \rangle \,\ge \, \langle Du,v \rangle ^{2k+1},$

(1)

with equality if and only if $\langle (DD^t)^k D u ,v \rangle = 0$, or there exists $\alpha > 0$ such that $Du = \alpha v$ and $D^t v = \alpha u$. This inequality extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy which asserts that if $D$ is a non-negative $n\!\times \!n$ symmetric matrix, and $u\!\in \!\mathbb{R}^n$ is a non-negative unit vector, then

$\langle D^k u, u \rangle \, \ge \,\langle Du,u \rangle ^{k},$

(2)

with equality, when $k\!\ge \!2$, if and only if $\langle D^k u, u \big \rangle = 0$, or there exists $\alpha \!>\!0$ such that $Du = \alpha u$. The generality of the inequality (1) derives not only from the fact that $D$ is not assumed to be symmetric or square, but from the fact that we admit two unit vectors $u$ and $v$ instead of the single unit vector $u$ appearing in inequality (2) of Blakley and Roy. We apply our result to verify the conjecture of A. Sidorenko (1993) in the non-symmetric case provided that the underlying graph is a path.