Wiman-Valiron theory in simply connected domains
The authors considered Wiman-Valiron theory for analytic functions in the unit disc \(\Delta(0,1)\) in their paper [J. Math. Anal. Appl. 367, No. 1, 137--145 (2010; Zbl 1205.30028)]. Here, they generalize this result to analytic functions in general simply-connected domains: Suppose that \(f\) is analytic in a simply-connected domain \(D\), which is not the whole plane, and let \(\omega\) be a univalent map of \(\Delta(0,1)\) onto \(D\). Let \(g=f(\omega)\) and suppose that \(g\) has order \(\rho>0\). Given \(\gamma\) satisfying \(0<\gamma<\rho/(2(\rho+1))\), let \(\zeta\) be such that \(|g(\zeta)|\geq N^{-\gamma} M(|\zeta|,g)\), where \(N=N(|\zeta|,g)\). There is a set \(E\subset (0,1)\) of lower density zero for which \[ \frac{N(r)}{-\log(1-r)} \to \rho+1 \] holds as \(r\to1\) outside \(E\); for every positive integer \(q\), \[ \frac{(\omega(\zeta)-\omega(0))^q f^{(q)}(\omega(\zeta))}{f(\omega(\zeta))}=(1+o(1))\left(\frac{(\omega(\zeta)-\omega(0))N}{\zeta \omega'(\zeta)}\right)^q \] as \(|\zeta|\to1\) outside \(E\); and \[ \frac{(\omega(\zeta)-\omega(0))N}{\zeta \omega'(\zeta)}\to\infty \] as \(|\zeta|\to1\) outside \(E\).
scientific article; zbMATH DE number 5965037
scientific article