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suivant: Preliminaries monter: A Bloch Constant for précédent: A Bloch Constant for


Since confirmation of the Bieberbach conjecture by de Branges, perhaps the outstanding open problem in complex analysis is that of finding the exact value of the Bloch constant.

Let $H(B)$ be the class of functions $w=f(z)$ holomorphic in the unit disc $B=\{z\in\mathbb {C}:\vert z\vert<1\}$. In 1925, Bloch [3] proved the famous theorem which bears his name:

Theorem 1 (Bloch)   There exists a positive constant $b$ such that if $f\in H(B)$ and $f^\prime(0)\neq
0$, then $f$ maps some subdomain of B biholomorphically onto a disc of radius $b\cdot\vert f^\prime(0)\vert$.

Such a disc is called a univalent disc for $f$. The Bloch constant may be described as:

\begin{displaymath}\beta=\mbox{inf}\{\beta_{f}:f\in H(B)\mbox{ with }f^\prime(0)=1\}\end{displaymath}

where $\beta_{f}=\mbox{sup}\{b: f(B)\mbox{ contains a univalent disc of radius
b}\}$. In this paper, we introduce Bloch constants for other classes of mappings and find one which is precisely equal to the classical Bloch constant $\beta$.

The following upper and lower estimates for $\beta$ were found by Lars Ahlfors and Grunsky [2] and Ahlfors [1]:


It is conjectured that the correct value of $\beta$ is precisely this upper bound. Recently, on the basis of Bonk's work [5] and the Schwarz-Pick lemma, Chen Huaihui and P. M. Gauthier [6] improved the lower bound for Bloch's constant further as follows:

\begin{displaymath}\frac{\sqrt{3}}{4}+2\cdot 10^{-4}\leq\beta.\end{displaymath}

Passing to several complex variables, a holomorphic mapping $f$ from a domain in $\mathbb {C}^{n}$ into $\mathbb {C}^{n}$ is said to be nondegenerate if $det\mathcal{J}_{f}$ is not identically zero on the domain. Let $B^{n}$ denote the open unit ball in $\mathbb {C}^{n}$. A nondegenerate mapping $f$ from $B^{n}$ into $\mathbb {C}^{n}$ is said to be normalized if $det\mathcal{J}_{f}(0)=1$, where $0$ denotes the origin in $\mathbb {C}^{n}$. For such $f$ we denote by $\beta_{f}$ the supremum of values $b$ such that the image $f(B^{n})$ contains a univalent ball of radius $b$. If we fix $K>0$ and consider the holomorphic mapping $f:\mathbb {C}^{2}
\longrightarrow\mathbb {C}^{2}$ defined by


then, $f$ is normalized but $\beta_{f}=1/\sqrt{K}$. Since $K$ can be chosen arbitrarily large, we see that there is no Bloch theorem for general holomorphic mappings, when $n>1$. One might argue that the correct generalization of the normalization $\lq\lq f^\prime(0)=1''$ to several variables is not $\lq\lq det
\mathcal{J}_{f}(0)=1''$. However, there are also examples of holomorphic mappings $f$, with the stronger normalization $\mathcal{J}_{f}(0)=I$ ($I$ is the identity mapping) and for which $\beta_{f}$ is arbitrarily small. One of the results of this article is to show that in the case of $\mathbb {C}^{2}$ the stronger normalization is correct, if the mapping $f$ satisfies the complexified Cauchy-Riemann equations. Thus, we see that for $n>1$ we need to restrict the class of mappings to a more specific subclass to obtain a Bloch theorem. One of the well known subclasses is the class of $K$-quasiregular mappings. For $n\geq 1$, let $\vert$ $\vert$ denote the usual norm in $\mathbb {C}^{n}$.

Definition 1 (Wu)   Let ${B}^{n}$ be the open unit ball of $\mathbb {C}^{n}$ and let $\mathcal{F}:{B}^{n}\longrightarrow\mathbb {C}^{n}$ be a family of holomorphic mappings. We say $\mathcal{F}$ is K-quasiregular iff there exists a constant $K$ so that, for each $f=(f_{1},\ldots,f_{n})$ of $\mathcal{F}$, the following holds throughout ${B}^{n}$,

...\mathcal{J}_{f}(z)\vert}^{1/n},\mbox{ for }\sigma=1,\ldots,n.

For such a class of mappings, it is possible to find a Bloch theorem in $\mathbb {C}^{n}$. Wu found a beautiful proof of this in [21]:

Theorem 2 (Wu, 1967)   Let $\mathcal{F}:{B}^{n}\longrightarrow\mathbb {C}^{n}$ be a K-quasiregular family of holomorphic mappings such that $\vert det \mathcal{J}_{f}$(0)$\vert$=1 for all $f$ $\in\mathcal{F}$. Then, there is a positive constant $c$ such that every $f\in\mathcal{F}$ possesses a univalent ball of radius $c$.

As Wu pointed out, this result also follows from the work of Bochner [4] in 1946. It is possible to find lower estimates for the Bloch constant for this class of mappings in [7], [17] and [20] provided we adapt correctly the definitions. We note also that the estimates depend on the $K$ of quasiregularity.

In this article, we use a generalization of complex numbers called bicomplex numbers ([13], [16], [14]) to find another subclass of mappings which has a Bloch constant in $\mathbb {C}^{2}$. Moreover, we find the estimates: $\frac{\beta}{\sqrt{2}}\leq\delta\leq\sqrt{2}\beta$ for this Bloch constant $\delta$, whenever our mappings are on the unit ball, and we find a specific domain of $\mathbb {C}^{2}$ where the Bloch constant has the same value as the Bloch constant for one variable. Finally we show that this class of mappings does not depend on the quasiregularity.
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suivant: Preliminaries monter: A Bloch Constant for précédent: A Bloch Constant for
Dominic Rochon