Subordination for BMOA (Q916832)

For every point a in the unit disc U, \(\psi_ a(z)=(z+a)/(1+\bar az)\) denotes the conformal map of U onto itself with \(\psi_ a(0)=a\). For a function f analytic in U, we set \(f_ a(z)=f(\psi_ a(z))-f(a)\), and the BMO norm \(B_ p(f)\) of f is defined by \(B_ p(f)=\sup \{\| f_ a\|_ p:\) \(a\in U\}\), where \(\| \cdot \|_ p\) denotes the ordinary Hardy norm. The family of all f for which \(B_ p(f)\) is finite is denoted by BMOA(U). It immediately follows from the subordination principle that the inequality \(B_ p(f\circ \psi)\leq B_ p(f)\) holds for every \(f\in BMOA(U)\) and every bounded analytic function \(\phi\) in U with \(| \phi (z)| <1\) for \(z\in U\), and from Ryff's theorem [\textit{J. V. Ruff}, Duke Math. J. 33, 347-354 (1966; Zbl 0148.302)] that the equality \(B_ p(f\circ \phi)=B_ p(f)\) holds for every \(f\in BMOA(U)\) if \(\phi\) is an inner function. In this paper, it is shown that there are fairly many nonlinear functions \(\phi\) such that the equality holds for every \(f\in BMOA(U)\). Such functions \(\phi\) are constructed by considering the composite function \(\psi\circ \lambda\) of any inner function \(\psi\) with a singularity at a point on the unit circle and a universal covering map \(\lambda\) of a domain D contained in U such that U-D is of positive logarithmic capacity.