Bloch constants in one and several variables

Ian Graham and Dror Varolin

Vol. 174 (1996), No. 2, 347–357

DOI: 10.2140/pjm.1996.174.347

Abstract

We give covering theorems in one variable for holomorphic functions on the unit disc with k-fold symmetry. In the case of convex maps we give a generalization, shown to us by D. Minda, to the case where a₂ = ⋯ = a_k = 0. In several variables we determine the Bloch constant (equivalently the Koebe constant) for convex maps of B_n with k-fold symmetry, k ≥ 2. We also estimate and in some cases compute the Bloch constant for starlike maps of B_n with k-fold symmetry. We compare the Bloch constant with the Koebe constant for such maps and determine values of n and k for which equality holds.

Mathematical Subject Classification 2000

Primary: 30C25

Secondary: 30C45, 32A30, 32H02

Milestones

Received: 28 December 1993

Published: 1 June 1996