  In this paper, we introduce a one parameter generalization of the famous
B\"ottcher-Wenzel (BW) inequality in terms of a $q$-deformed commutator. For $n
\times n$ matrices $A$ and $B$, we consider the inequality \[
\Re\langle[B,A],[B,A]_q\rangle \le c(q) \|A\|^2 \|B\|^2, \] where $\langle A,B
\rangle = {\rm tr}(A^*B)$ is the Hilbert-Schmidt inner product, $\|A\|$ is the
Frobenius norm, $[A,B] =AB-BA$ is the commutator, and $[A,B]_q =AB-qBA$ is the
$q$-deformed commutator. We prove that when $n=2$, or when $A$ is normal with
any size $n$, the optimal bound is given by \[ c(q) = \frac{(1+q)
+\sqrt{2(1+q^2)}}{2}. \] We conjecture that this is also true for any matrices,
and this conjecture is perfectly supported for $n$ up to $15$ by numerical
optimization. When $q=1$, this inequality is exactly BW inequality. When $q=0$,
this inequality leads the sharp bound for the $r$-function which is recently
derived for the application to universal constraints of relaxation rates in
open quantum dynamics.
