Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in [formula omitted

Abstract

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for a.e. t≥0, and which satisfies a natural assumption. Next, we prove that if f∈M(Bn), then the nonlinear resolvent family {Jr}r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector field is of divergence type, where Jr=Jr[f]=(In+rf)−1, r≥0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1+r)Jr can be embedded as the first element of a normal Loewner chain and the family {(1+r)Jr[f]:0≤r<∞,f∈M(Bn)} is a compact subset of S0(Bn). Also, we deduce that the shearing of (1+r)Jr (r≥0) associated with the family M(B2) is quasi-convex of type A and also starlike of order 4/5. We give a sufficient condition for (1+r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of Cn onto itself. Finally, sharp coefficient bounds for the nonlinear resolvent families of certain subsets of M(Bn), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2, will be obtained.