Suppose that is a power series of the form Then for any r, 0 < r < 1, has zeros in .
We next consider bounds on the size of . Since for , it suffices to consider .
Suppose that z satisfies |z| < 1 and that for some power series of the form . Then and , with equality if and only if and . Furthermore, there exists a such that if , then z is a negative real number.
The argument presented above is inefficient, and shows only that some value of is allowable. 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. Numerical evidence suggests that the minimal value of |z| over is about 0.734. The method of Section 6 can be used to obtain estimates for the minimal value of |z| over that are as accurate as one desires.
By Proposition 3.1 of the next section, . Since is stable under and closed, it follows that .
In  it was shown that implies . 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 find that for z =1-x +iy with x and y small, x > 0, if then fails, so . We next show that there are points in W which approach 1 along trajectories tangent to the real axis.
There exists a sequence of points such that as and