Suppose that $A$ is an $n \times n$ matrix. Let $\text{W}$ denote the numerical range which is given by \begin{align*} \text{W}(B) = \{ \langle v, B v \rangle \mid \vert \vert v \vert \vert = 1 \}. \end{align*} Suppose that I know $\text{W}(\text{Im}(A))$ and $\text{W}(\text{Re}(A))$, but I do not know $\text{W}(A) $.
It follows that $\text{W}(A) \subset \text{W}(\text{Re}(A)) + i \text{W}(\text{Im}(A))$, but can I conclude any lower bound?