Numerical Ranges and Spectral Sets: the unbounded case

It is known that, if $\mathrm{\Omega }$ $\subset$ C is a convex set containing the numerical range of an operator A, then $\mathrm{\Omega }$ is a C $\mathrm{\Omega }$ -spectral set for A with C $\mathrm{\Omega }$ $\le$ 1+ \sqrt 2. We improve this estimate in unbounded cases.