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

Abstract

Let Y be a complex Banach space and let $$r\ge 1$$. In this paper, we are concerned with an extension operator $$\varPhi _{\alpha , \beta }$$ that provides a way of extending a locally univalent function f on the unit disc $$\mathbb {U}$$ to a locally biholomorphic mapping $$F\in H(\varOmega _r)$$, where $$\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: |z_1|^2+\Vert w\Vert _Y^r 0$$, $$\zeta \in \mathbb {U}$$, then $$F =\varPhi _{\alpha , \beta }(f)$$ can be embedded as the first element of a g-Loewner chain on $$\varOmega _r$$, for $$\alpha \in [0, 1]$$, $$\beta \in [0, 1/r]$$, $$\alpha +\beta \le 1$$. We also show that normalized univalent Bloch functions on $$\mathbb {U}$$ (resp. normalized uniformly locally univalent Bloch functions on $$\mathbb {U}$$) are extended to Bloch mappings on $$\varOmega _r$$ by $$\varPhi _{\alpha ,\beta }$$, for $$\alpha >0$$ and $$\beta \in [0,1/r)$$ (resp. for $$\alpha =0$$ and $$\beta \in [0,1/r]$$). In the case of the Muir type extension operator $$\varPhi _{P_k}$$, where $$k\ge 2$$ is an integer and $$P_k:Y\rightarrow \mathbb {C}$$ is a homogeneous polynomial mapping of degree k with $$\Vert P_k\Vert \le d(1,\partial g(\mathbb {U}))/4$$, we prove a similar extension result for the first elements of g-Loewner chains on $$\varOmega _k$$. Next, we consider a modification of the Muir type extension operator $$\varPhi _{G,k}$$, where $$k\ge 2$$ is an integer and $$G:Y\rightarrow \mathbb {C}$$ 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 $$\mathbb {U}$$ such that $$g(0)=1$$, $$\mathfrak {R}g(\zeta )>0$$, $$\zeta \in \mathbb {U}$$, and g satisfies a natural boundary condition, and if the extension operator $$\varPhi _{G,k}$$ maps g-starlike functions from the unit disc $$\mathbb {U}$$ into starlike mappings on $$\varOmega _k$$, 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 $$\mathbb {U}$$ to Bloch mappings on $$\varOmega _k$$ by $$\varPhi _{P_k}$$.