CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR

doi:10.1007/s10473-019-0612-9

Abstract

Abstract: Let λ_G(z)|dz|be the hyperbolic metric on a simply connected proper domain G ⊂ C containing the origin, and let||·||_j be the Banach norms of C^n_j for j=1, 2, …, k.This note is to prove that if f is a normalized biholomorphic convex function on G, then
Φ_{N,1/p1,…,1/pk}(f)(z)=F_{1/p1,…,1/pk}(z)=f(z₁), (f'(z₁))^1/p1z, …, (f'(z₁))^1/pkw)
is a normalized biholomorphic convex mapping on
Ω_N={(z₁, z, …, w) ∈ C×C^n₁×…×C^n_k:||z||₁^p1 + … +||w||_k^pk<1/λ_G (z₁)},
where N=1 + n₁ + … + n_k and the branch is chosen such that (f'(z₁))^1/p_j|_z₁=0=1, j=1, …, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.

Key words: Roper-Suffridge operator, convex mapping, hyperbolic metric

CLC Number:

32H02

Danli ZHANG, Huiming XU, Jianfei WANG. CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR[J].Acta mathematica scientia,Series B, 2019, 39(6): 1619-1627.

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