A characterization of Möbius transformations (Q5891473)

It is shown that if \(\theta\) is an inner function in \(H^\infty(\mathbb D)\) whose derivative \(\theta'\) belongs to the Nevanlinna class \(N\), then \(\theta'\) is an outer function if and only if \(\theta\) is a Möbius transformation. Using a result from a previous paper [Comput. Methods Funct. Theory 13, No. 3, 449--457 (2013; Zbl 1291.30254)], the author also shows that if \(f\) belongs to the unit ball of \(H^\infty(\mathbb D)\) and nonconstant, then \(f\) is a Möbius transformation if and only if there is an increasing function \(H: ]0,\infty[\to ]0,\infty[\) with \(\lim_{t\to\infty} H(t)=\infty\) such that NEWLINE\[NEWLINE H\left(\frac{1-|f(z)|^2}{1-|z|^2)}\right)\leq |f '(z)|.NEWLINE\]