On the composition of functions of bounded mean oscillation with multivalent analytic functions (Q913986)

Let \(\Omega\) be a plane domain. Denote by X(\(\Omega\)) the space of locally integrable functions g on \(\Omega\) which satisfy the BMO-type condition \[ \sup (1/m(B))\int_{B}| g-g(B)| dm<\infty, \] where dm denotes 2-dimensional Lebesgue measure, g(B) the average of g on B, and the sup is over all disks \(B\subset \Omega\). The main result characterizes analytic functions f from the unit disk D into \(\Omega\) with the property that \(g\in X(\Omega)\) implies \(g\circ f\in X(D)\) as precisely the ones for which there exist a constant \(\alpha >0\) and integer p such that f is p-valent in every hyperbolic disk of radius \(\alpha\). The special case when f is a universal covering map was proved in a previous paper by the author. Results of this type when f is a quasiconformal homeomorphism are due to H. M. Reimann and to P. W. Jones. A corollary of the present work is that composition with an entire function f preserves X(\({\mathbb{C}})\) if and only if f is a polynomial. The proof of the main theorem uses the following result: If \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) is p-valent in D and locally univalent, then \(| a_ j| \leq C(p)\) for \(j=2,...,p\). The author supplies a proof. The result follows also from work of \textit{Ch. Pommerenke} [Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155, 108-154 (1964; Zbl 0128.301), Satz. 1.3, p. 119].