On removal of periods of conjugate functions in multiply connected domains
Let G be a domain in the complex plane bounded by n analytic Jordan curves \(\gamma_ 1,...,\gamma_ n\), \(\Gamma =\cup \gamma_ i\). The author proves that for each (n-1)-tuple of real numbers \(\Lambda_ 1,...,\Lambda_{n-1}\) and each positive Borel measure \(\mu\) on \(\Gamma\) satisfying \(\mu (\gamma_ i)>0\), \(i=1,...,n-1\) there exist real (real positive) \(\lambda_ 1,...,\lambda_{n-1}\) \((\lambda_ 1,...,\lambda_ n)\) such that the periods of the conjugate harmonic functions defined by the Poisson integral of the measure \({\tilde \mu}\): \({\tilde \mu}|_{\gamma_ i}\equiv \lambda_ i\mu |_{\gamma_ i}\) along \(\gamma_ 1,...,\gamma_{n-1}\) are equal to \(\Lambda_ 1,...,\Lambda_{n-1}.\) Some applications of this result (such as the existence in multiply connected domains of an analytic function in the Nevanlinna or the Hardy classes with prescribed modulus of boundary values) are given too.
periods of the conjugate harmonic functions
scientific article; zbMATH DE number 3950963