Bounding the commutator [A,B] in terms of the numerical radius Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$
The answer is known at least in the following cases:


*

*the operator norm $\|A\|_2=\sup\frac{\|Ax\|_2}{\|x\|_2}$ where the norm over $\mathbb C^n$ is the standard Hermitian $\|x\|_2^2=\sum_j|x_j|^2$. Then
$$\|[A,B]\|_2\le2\|A\|_2\|B\|_2$$
is optimal for $n\ge2$.

*the Frobenius norm $\|A\|^2_F=\sum_{i,j}|a_{ij}|^2$. Then a theorem by Böttcher & Wentzel (2008) tells us that
$$\|[A,B]\|_F\le\sqrt2\|A\|_F\|B\|_F,$$
and again this is optimal.



I have a third norm in mind, yet of a different nature: the numerical radius
  $$r(A)=\sup_{x\ne0}\frac{|x^*Ax|}{\|x\|^2}.$$
  This is the smallest radius of a disk $D(0;r)$ containing the numerical range (or Hausdorffian) of the matrix.
  What is the optimal constant $C_{nr}$ such that $r([A,B])\le C_{nr}r(A)r(B)$ for all $A,B$ in ${\bf M}_n(\mathbb C)$ ?

Let me point out that $r$ is not submultiplicative. We have at best $r(MN)\le 4r(M)r(N)$, which gives by the triangle inequality $r([A,B])\le8r(A)r(B)$, but this is certainly not optimal. However, it is a super-stable norm, in the sense that $r(M^k)\le r(M)^k$ for every $k\ge1$.
This question naturally extends to $n$-commutators, in the spirit of my previous question Standard polynomials applied to matrices .
Edit. See below Piotr Migdal's answer and my adaptation of it. It gives $C_{nr}=4$.
 A: I got
$$r([A,B])\leq 4\sqrt{2} r(A) r(B).$$
It is lower than $8$ but still higher than the conjuncture $C_{nr}=4$.  
I used the following facts: 


*

*For normal (i.e. $X^\*X=XX^\*$) matrices we have $r(X)=\sigma_1(X)$ (the largest singular value of X).

*$\sigma_1(XY-YX)\leq 2 \sigma_1(X)\sigma_1(Y)$

*Also, note that for $X$ and $Y$ hermitian (i.e. $X^\*= X$ and $Y^\* = Y$ ) we have
$$r(X+iY) \geq \max\left(\sigma_1(X),\sigma_1(Y)\right),$$
$$r(X+iY) \leq \sqrt{\sigma_1^2(X) + \sigma_1^2(Y)}.$$
Lets decompose $A$ and $B$ in their hermitian and antihermitian parts,
$$A = A_h + i A_a, \quad B= B_h+i B_a.$$
Then 
$$r^2([A,B]) \leq \left( \sigma_1([A_h,B_h]-[A_a,B_a])\right)^2
+ \left( \sigma_1([A_h,B_a]+[A_a,B_h])\right)^2$$
$$\leq\left( 2\sigma_1(A_h)\sigma_1(B_h)+2\sigma_1(A_a)\sigma_1(B_a) \right)^2
+ \left( 2\sigma_1(A_h)\sigma_1(B_a)+2\sigma_1(A_a)\sigma_1(B_h) \right)^2$$
$$=4 ( \sigma_1^2(A_h) + \sigma_1^2(A_a) )( \sigma_1^2(B_h) + \sigma_1^2(B_a) )
+16 \sigma_1(A_h) \sigma_1(A_a) \sigma_1(B_h) \sigma_1(B_a)$$
$$\leq 8( \sigma_1^2(A_h) + \sigma_1^2(A_a) )( \sigma_1^2(B_h) + \sigma_1^2(B_a) )$$
$$\leq 32 r^2(A)r^2(B).$$
A: Here is a partial result. 
Claim. $4 \le C_{nr} \le 8$.
Proof. The upper bound has already been shown by the OP. The lower-bound follows by 
\begin{equation*}
  A =
  \begin{bmatrix}
    0 & 1\\\\
    0 & 0
  \end{bmatrix},\qquad
  B =
  \begin{bmatrix}
    0 & 0\\\\
    -1 & 0
  \end{bmatrix}
\end{equation*}
for which 
$$\frac{r([A,B])}{r(A)r(B)} = \frac{1}{\frac 12\times\frac 12}=4.$$
(Note: Slightly more generally, the $1$ in the above matrices can be replaced by an nonzero scalar).
Based on some experiments mentioned in my comments to Denis, I am led to the following attractive conjecture.
Conjecture: $C_{nr}=4$.

Additional Remarks. 
Define $$X :=
  \begin{bmatrix}
    0 & 1\\\\
    0 & 0
  \end{bmatrix},$$
and let $A$, $B$ be arbitrary. Then, it is easy to see that we have the commutator inequality:
\begin{equation*}
  r(X \otimes [A,B]) \le 4 r(X \otimes A) r(X \otimes B), 
\end{equation*}
where $\otimes$ denotes the Kronecker product.
A: The answer by Piotr Migdal can be modified to give the accurate inequality 
$$r([A,B])\le4r(A)r(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$
The only new argument is that for every matrix $M$, there exists an angle $\theta$ such that $r(M)=\|{\rm Re}(e^{-i\theta}M)\|_2.$ 
Actually, we do have
$$r(M)=\sup_\alpha\|{\rm Re}(e^{-i\alpha}M)\|_2.$$
Hereabove, the real part is defined as ${\rm Re} N=\frac12(N+\bar N^T)$. Notice that I employ the notation $\|\cdot\|$ (operator norm) which coincides with $\sigma_1$.
Let us apply this to $M=[A,B]$. With $\theta$ as above, let us decompose $e^{-i\theta}A=A_{\theta h}+iA_{\theta a}$. Then let us proceed as Piotr did:
$$r([A,B])=\|{\rm Re}[e^{-i\theta}A,B]\| = \|i[A_{\theta h},B_a]+i[A_{\theta a},B_h]\|\le2(\|A_{\theta h}\|\cdot\|B_a\|+\|A_{\theta a}\|\cdot\|B_h\|),$$
which gives 
$$r([A,B])\le4r(e^{-i\theta}A)r(B).$$
Hence the result.
