By Sheng Gong

This booklet offers with the speculation of convex and starlike biholomorphic mappings in different complicated variables. The underlying subject is the extension to a number of complicated variables of geometric points of the classical conception of univalent features. this can be the 1st ebook which systematically experiences this subject. It gathers jointly, and provides in a unified demeanour, the present situation for convex and starlike biholomorphic mappings in numerous complicated variables. nearly all of the implications provided are as a result of writer, his co-workers and his scholars. *Audience:* This quantity should be of curiosity to analyze mathematicians whose paintings consists of a number of advanced variables and one complicated variable.

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**Sample text**

3 The criterion for starlikeness for holomorphic mappings on r-domains In this section, we· will consider the criteria for starlikeness for holomorphic mappings on r-domains. Let n c en be a domain. in. We will prove the following three theorems in this section. (Gong, Wang and Yu [3], [6]). 1 Let n c en be an r-domain, and let f : n -+ en be a locally biholomorphic mapping on n, which satisfies the following conditions 1. r(p) = 0, f(p) = 0 where p is a fixed point in n, and r(z) > 0 when z E n\{p}; 2.

E. a harmonic function on 3.. Hence ~e{h(e)} ~ a when E~. 1), since op~~z) e = ,\ o~~) when ,\ E [0,00). Case 2. The point Zo is an essential boundary point of n. 5 in Krantz [1], there exists a neighborhood of Zo in an which consists of essential boundary points of n. Choosing a neighborhood of Zo on an sufficiently small, and joining each point in this neighborhood to the origin, we obtain a cone type domain c en, which is a domain of holomorphy. For any fixed t E (0,1), r E [0,1], let n C) = f-l«l - r)f(teZo)) CE A 'Pr,t (c,.

1, f is starlike with respect to the origin on n. 1. 1. 1. Let r(z) = l~~{Z) where p(z) is the Minkowski functional of n. Then r(z) is a continuous exhaustion function on n, hence n is a r-domain. 1). 1 are satisfied if we let p = o. 1. 2. Let zEn, ~ ,X ~ 1. 2). 3). If r{z) E C 1 except for a lower dimensional manifold, then for all ,X E (O, 1) we have r{z) - r{z~) 1-'x ~ 0. J iJ i i = (WI'···' Wn ) = f. 5). 2. 2. Let r(z) = Cn(O,z). The set {z En: Cn(O,z) < a} is relative compact in n, since n is a Caratheodory complete domain.