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An extremum Problem related to Morera’s theorem. (English. Russian original) Zbl 0899.30002
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 $$a$$ $$R(T)= a\sqrt 3/2$$.
##### MSC:
 30A05 Monogenic and polygenic functions of one complex variable
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