where . In this paper, we introduce Bloch constants for other classes of mappings and find one which is precisely equal to the classical Bloch constant . The following upper and lower estimates for were found by Lars Ahlfors and Grunsky [2] and Ahlfors [1]:

It is conjectured that the correct value of 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:

Passing to several complex variables, a holomorphic mapping from a domain in into is said to be nondegenerate if is not identically zero on the domain. Let denote the open unit ball in . A nondegenerate mapping from into is said to be normalized if , where denotes the origin in . For such we denote by the supremum of values such that the image contains a univalent ball of radius . If we fix and consider the holomorphic mapping defined by

then, is normalized but . Since can be chosen arbitrarily large, we see that there is no Bloch theorem for general holomorphic mappings, when . One might argue that the correct generalization of the normalization to several variables is not . However, there are also examples of holomorphic mappings , with the stronger normalization ( is the identity mapping) and for which is arbitrarily small. One of the results of this article is to show that in the case of the stronger normalization is correct, if the mapping satisfies the complexified Cauchy-Riemann equations. Thus, we see that for 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 -quasiregular mappings. For , let denote the usual norm in .