An indestructible Blaschke product in the little Bloch space (Q1325628)

An infinite Blaschke product \(B\) is constructed so that \(B(z)- a/(1- \bar a B(z))\) is in the little Bloch space \({\mathcal B}_ 0\) for all complex \(a\), \(| a|< 1\). This answers in the affirmative a question posed by K. Stephenson. Also, a VMOA function \(f: \| f\|_{H^ \infty}= 1\) is constructed whose range set \(R(f,a):= \{w: \exists z_ n\to a,\;f(z_ n)= w\}\) at each point \(a\) on the unit circle equals the whole open unit disk.