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 $\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: |z_1|^2+\Vert w\Vert _Y^r<1\}$. We prove that if f can be embedded as the first element of a g-Loewner chain on $\mathbb {U}$, where g is a convex (univalent) function on $\mathbb {U}$ such that $g(0)=1$ and $\mathfrak {R}g(\zeta )>0$, $\zeta \in \mathbb {U}$, 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 $\varPhi _{P_k}$.