An extremum Problem related to Morera’s theorem. (English. Russian original) Zbl 0899.30002

Volchkov, V. V.

Math. Notes 60, No. 6, 605-610 (1996); translation from Mat. Zametki 60, No. 6, 804-809 (1996).

The following extension of Morera’s theorem has been proved by C. Berenstein and R. Gay [J. Anal. Math. 52 133-166 (1989; Zbl 0668.30037)]: Let \(T \subseteq B_r= \{z\in\mathbb{C}:| z| <r\}\) be a given triangle, \(f\) be a continuous function in \(B_R\), and \(2r\leq R\). Then \(f\) is holomorphic in \(B_R\) iff \[ \int_{\partial (\sigma T)} f(z)dz=0, \quad \forall \sigma\in M: \sigma T\subseteq B_R. \tag{1} \] Here \(M\) is the group of Euclidean motions of the complex plane \(\mathbb{C}\). The present author has studied the problem of finding the minimum value \(R=R(T)\) such that for any function \(f\in C(B_R)\) condition (1) implies that \(f\) is holomorphic and has found the value for squares and half-disks. In the article under review the author proves that for regular triangles \(T\) of side \(a\) \(R(T)= a\sqrt 3/2\).

Reviewer: A.I.Kheyfits (Brooklyn)

MSC:

30A05

Monogenic and polygenic functions of one complex variable

Keywords:

description of holomorphic functions; Morera’s theorem