Herman Rings Of Blaschke Products Of Degree 3

Let F(a,lambda) be the Blaschke product of the form F(a,lambda) = lambda z(2)((z - a)/(1 - (a) over barz)) and alpha denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any a there exists a constant a(0) >= 3 and a continuous function lambda(a) such that F(a,lambda) (a) possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a(0). It is remarked that the question whether a(0) = 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a(0) is strictly larger than 3.