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Some uniqueness theorems concerning holomorphic mappings. (English) Zbl 0755.32002

Let \(D\) be \(T_ n=\{z=(z_ 1,\ldots,z_ n)\in\mathbb{C}^ n:| z_ j|<1\) for \(j=1,\ldots,n\}\) or \(B_ \alpha=\{z\in\mathbb{C}^ n:\sum^ n_{j=1}| z_ j|^ \alpha<1\}\), where \(\alpha=(\alpha_ 1,\dots,\alpha_ n)\in\mathbb{R}^ n_ +\). The paper is primarily concerned with certain conditions under which some power series (about 0) which represent a holomorphic map \(f\) of \(D\) reduce themselves to monomials. The motivation comes from the classical case in which \(f=P_ k\), when \(D=T_ 1\), \(f(D)\subseteq D\) and \(P_ k\) is one of the terms of \(f\) (so that for some \(a_ k\in\mathbb{C}\), \(P_ k(z)=a_ kz^ k\) for all \(x\in D)\) and the term maps \(D\) onto \(D\) (so that \(| a_ k|=1\), \(k>0)\). The paper has five theorems. Theorem 1 extends (easily) the classical result to the case when \(D=T^ n\). Here (to typically illustrate) \(f=(f_ 1,\dots,f_ n)\), \(f_ s=\sum^{+\infty}_{m=0}P_ m^{[s]}\) on \(D\) where \(P_ m^{[s]}\) is a homogeneous polynomial of degree \(m(m=0,1,\dots,s=1,\dots,n)\) and \(P_ k=(P^{[1]}_{k_ 1},\dots,P^{[n]}_{k_ n})\). It is shown that \(f=P_ k\), under the (frequent) assumptions that \(k\in\mathbb{R}^ n_ +\) and \(P_ k(z)=(z^{k_ 1}_ 1,\dots,z_ n^{k_ n})\) for all \(x\in D\). Theorem 2 extends Schwarz’s lemma to the case when \(f\) maps \(RT_ 1\) into \((1/M)B_ \alpha\) where \(R,M\in\mathbb{R}_ +\). Theorem 3 and 4 consider the case when \(f\) maps \(B_ \alpha\) into \(B_ \gamma\) where \(\gamma=(\alpha_ 1/k_ 1,\dots,\alpha_ n/k_ n)\). The assumption of \(f\) “starting” with \(P_ k\) in Theorem 3 is replaced by the holomorphy of \(f\) on \(\overline B_ \alpha\) in Theorem 4. Theorem 5 is an easy consequence of Theorem 3 and 4 through applying a non-singular linear transformation. [Some apparent misprints may have to be negociated].

MSC:

32A05 Power series, series of functions of several complex variables
32A10 Holomorphic functions of several complex variables
30B10 Power series (including lacunary series) in one complex variable
32A30 Other generalizations of function theory of one complex variable
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