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 Phi(alpha,beta) that provides a way of extending a locally univalent function f on the unit disc U to a locally biholomorphic mapping F is an element of H(Omega(r)), where Or = {(z(1), w) is an element of CxY : vertical bar z(1)vertical bar(2) + vertical bar vertical bar w vertical bar vertical bar(r)(Y) < 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(zeta) > 0,zeta is an element of U, then F = Phi(alpha,beta) (f) can be embedded as the first element of a g-Loewner chain on Omega(r), for alpha is an element of [0, 1], beta is an element of [0, 1/r], alpha + beta <= 1. We also showthat normalized univalent Bloch functions on U(resp. normalized uniformly locally univalent Bloch functions on U) are extended to Bloch mappings on Or by Phi(alpha,beta), for a > 0 and ss. [0, 1/r) (resp. for a = 0 and ss. [0, 1/r]). In the case of the Muir type extension operator Phi(Pk), where k >= 2 is an integer and P-k : Y -> C is a homogeneous polynomial mapping of degree k with vertical bar vertical bar P-k vertical bar vertical bar = d(1, partial derivative g(U))/4, we prove a similar extension result for the first elements of g-Loewner chains on Ok. Next, we consider a modification of the Muir type extension operator Omega(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, Tg(zeta) > 0, zeta is an element of U, and g satisfies a natural boundary condition, and if the extension operator Omega(G),(k) maps g-starlike functions from the unit disc U into starlike mappings on Ok, 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 Omega(k) by Phi(Pk)