g -Loewner chains, Bloch functions and extension operators in complex Banach spaces

Let Y be a complex Banach space and let r≥ 1 . In this paper, we are concerned with an extension operator _α , β that provides a way of extending a locally univalent function f on the unit disc 𝕌 to a locally biholomorphic mapping F∈ H( _r) , where _r={(z_1,w)∈ℂ× Y: |z_1|^2+‖ w‖ _Y^r<1} . We prove that if f can be embedded as the first element of a g -Loewner chain on 𝕌 , where g is a convex (univalent) function on 𝕌 such that g(0)=1 and ℜg(ζ )>0 , ζ∈𝕌 , then F = _α , β(f) can be embedded as the first element of a g -Loewner chain on _r , for α∈ [0, 1] , β∈ [0, 1/r] , α +β≤ 1 . We also show that normalized univalent Bloch functions on 𝕌 (resp. normalized uniformly locally univalent Bloch functions on 𝕌 ) are extended to Bloch mappings on _r by _α ,β , for α >0 and β∈ [0,1/r) (resp. for α =0 and β∈ [0,1/r] ). In the case of the Muir type extension operator _P_k , where k≥ 2 is an integer and P_k:Y→ℂ is a homogeneous polynomial mapping of degree k with ‖ P_k‖≤ d(1,∂ g(𝕌))/4 , we prove a similar extension result for the first elements of g -Loewner chains on _k . Next, we consider a modification of the Muir type extension operator _G,k , where k≥ 2 is an integer and G:Y→ℂ is a holomorphic function such that G(0)=0 and DG(0)=0 , and prove that if g is a univalent function with real coefficients on 𝕌 such that g(0)=1 , ℜg(ζ )>0 , ζ∈𝕌 , and g satisfies a natural boundary condition, and if the extension operator _G,k maps g -starlike functions from the unit disc 𝕌 into starlike mappings on _k , then G must be a homogeneous polynomial of degree at most k . We also obtain a preservation result of normalized uniformly locally univalent Bloch functions on 𝕌 to Bloch mappings on _k by _P_k .