Meromorphic univalent functions in an annulus (Q1117343)

Let F(p) denote the class of meromorphic univalent functions in the annulus \(1<| z| <R\) with pole at \(p\in (1,R)\), which satisfy \(| f(z)| \to 1\) as \(| z| \to 1\) and \(| f(z)| >1\) for \(| z| >1\). The authors prove that for every convex increasing function \(\phi\), for \(r\in (1,R)\), and for \(f\in F(p)\), \[ \int^{\pi}_{-\pi}\phi (\pm \log | f(re^{i\theta})|)d\theta \leq \int^{\pi}_{-\pi}\phi (\pm \log | g_ p(re^{i\theta})|)d\theta, \] where \(g_ p\in F(p)\) maps the annulus onto the exterior of the unit disk with a certain interval deleted. For analytic univalent functions in an annulus the analogue of this theorem was proved by the reviewer (1974), and for a different class of meromorphic functions by Kirwan and Schober (1976).