A Generalization of the B\"{o}ttcher-Wenzel inequality for three rectangular matrices

Let $m,n$ be positive integers. For all $m\times n$ complex matrices $A, C$ and an $n\times m$ matrix $B$, we define a generalized commutator as $ABC-CBA$. We estimate the Frobenius norm of it, and finally get the inequality, which is a generalization of the B\"{o}ttcher-Wenzel inequality. If $n=1$ or $m=1$, then the Frobenius norm of $ABC-CBA$ can be estimated with a tighter upper bound.