We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.

Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire function. It is easy to see that a mapping theorem for the numerical range in the sense of $W(f(A)) = f(W(A))$ does not hold, in general. The following example gives a bit more insight:

**Example.** Let $A$ be the $3\times 3$-diagonal matrix with diagonal entries $0, \pi i, 2\pi i$ and let $f(z) = e^z$. Then $W(f(A))$ is the line segment $[-1,1]$, but $f(W(A))$ is the complex unit circle. In fact, this even shows that...

we don't have $W(f(A)) = \operatorname{conv}(f(W(A)))$, in general (where $\operatorname{conv}$ denotes the convex hull).

we don't have an inclusion theorem of the kind $W(f(A)) \subseteq f(W(A))$, in general.

Now, it seems natural to ask:

**Question.** Is there also a counterexample known for the more general inclusion
\begin{align*}
W(f(A)) \subseteq \operatorname{conv}(f(W(A))) \qquad (*)
\end{align*}
or is it an open problem whether $(*)$ holds?

**Note.** It is certainly *not known that $(*)$ is true*, due to the following reasoning:

(i) If $(*)$ is true, this immediately implies $w(f(A)) \le \sup_{z \in W(A)} \lvert f(z) \vert$, where $w$ denotes the numerical radius.

(ii) By the well-known inequality $\|B\| \le 2w(B)$ for every matrix $B$ (where $\|\,\cdot\,\|$ denotes the norm induced by the $2$-norm on $\mathbb{C}^n$), $(*)$ would thus imply that Crouzeix's conjecture \begin{align*} \|f(A)\| \le 2 \sup_{z \in W(A)} \lvert f(z) \vert \end{align*} is true.

**Remark.** It is easy to see that $(*)$ holds for every normal matrix (which also explains why it is true in the above example), but I could not even figure out whether it holds for $2 \times 2$ Jordan blocks.

**Disclaimer.** I am, of course, not asking whether Crouzeix's conjecture is true. I am asking whether the more general assertion $(*)$ is known to be false or whether it is an open problem.