On an extremum problem for Bloch functions (Q2367331)

Suppose that \(f\) is a Bloch function, that is, \(f\) is analytic in the open unit disk and \(M(r)= \sup_{| z|<1} (| f'(z)| (1- | z|^ 2))<+\infty\). Let \(L(f)= \{z\): \(| z|<1\) and \(| f'(z)| (1-| z|^ 2)= M(r)\}\). Suppose that \(L(f)\) contains a sequence \(S\) of distinct points having an accumulation point in the open unit disk. It is proved that if \(S\) is contained in some circle \(C\) then \(L(f)=C\cap D\). Also if \(S\) is contained in some straight line \(L\) then \(L(f)=L\cap D\). Moreover, the functions \(f\) are determined in either the case of a circle or of a line.