The minimum modulus of Blaschke products (Q1079986)

For a given Blaschke product B, let \(m(r,B)=\inf_{| z| =r}| B(z)|\). The authors show that (1-\(| z|)\log | B(z)|\) has fine limit 0 everywhere on the boundary of the unit disc, and that for some subset \(\Delta\) (0,1), with its complement minimally thin at 1, (1-r)log m(r,B)\(\to 0\) as \(r\to 1\) on \(\Delta\). This result is used to extend identity theorems of Shaginyan and Heins, and to supplement a theorem of Gol'dberg on meromorphic functions with logarithmic derivative of bounded characteristic.