With a little more care one could show by an extension of the method used above that is allowable, so that any with is real. In Section 6 we present a variation of this method that uses machine computation instead of careful estimates to establish rigorously that is allowable. By Proposition 3.1 of the next section, . Since is stable under and closed, it follows that . Theorem 2.1 immediately leads to the bound for .
Numerical evidence suggests that for .
There are with . The methods outlined in Section 6 can be used to obtain precise bounds. We can analyze inequality for z close to 1. We next show that there are points in W which approach 1 along trajectories tangent to the real axis.