Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$.
It is well know that if $A$ is hyponormal operator, then $$r(A)=\omega(A)=\|A\|,$$ where $r(A)$, $\omega(A)$ and $\|A\|$ denote respectively the spectral radius, the numerical radius and the norm of $A$.
I want to find an example of an operator $A$ such that the equality $$r(A)=\omega(A)=\|A\|,$$ does not hold.
$$A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ on the Hilbert space $\mathbb{C}^2$ with the Euclidean inner product has
$$r(A)=1 \\ \omega(A)=\max_{|x|^2+|y|^2=1} |x|^2+y\overline{x}+|y|^2=3/2 \\ \| A \|=\max_{|x|^2+|y|^2=1} \sqrt{|x+y|^2+|y|^2}=\frac{\sqrt{5}+1}{2}.$$
Note that regardless of any assumption you have $r(A) \leq \omega(A) \leq \| A \|$, and this example shows that both inequalities can simultaneously be strict.