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On Cauchy-Riemann conditions in the class of functions with summable module and some boundary properties of analytic functions. (Russian) Zbl 0593.30002

The following generalization of the Looman-Menchoff theorem is obtained: Let G be a domain in the complex plane \({\mathbb{C}}\) and \(f: G\to {\mathbb{C}}\), \(f=u+iv\) be a function subject to the conditions listed below: 1. Derivatives \(u_ x\), \(u_ y\), \(v_ x\), \(v_ y\) exist in \(G\setminus E\), where \(E=\cup^{\infty}_{n=0}E_ n\) and \(\{E_ n\}\) are closed sets with finite one dimensional Hausdorff measures. 2. The functions u and v are linearly continuous in G and satisfy the Cauchy-Riemann system almost everywhere. (A function \(g: G\to {\mathbb{R}}\) is linearly continuous at (a,b)\(\in G\), if the functions \(x\to g(x,b)\) and \(y\to g(a,y)\) are continuous at the points a and b respectively.) 3. The function \(z\to | f(z)|\) belongs to \(L^ 1_{loc}(G)\). Then f is analytic in G.
Reviewer: T.Genchev

MSC:

30A05 Monogenic and polygenic functions of one complex variable
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