Geodesic
Morera-type theorems in the hyperbolic disc
Izvestiya. Mathematics , Tome 82 (2018) no. 1, pp. 31-60

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Let $G$ be the group of conformal automorphisms of the unit disc $\mathbb{D}=\{z\in\mathbb{C}\colon |z|1\}$. We study the problem of the holomorphicity of functions $f$ on $\mathbb{D}$ satisfying the equation $$ \int_{\gamma_{\varrho}} f(g (z))\, dz=0 \quad \forall \, g\in G, $$ where $\gamma_{\varrho}=\{z\in\mathbb{C}\colon |z|=\varrho\}$ and $\rho\in (0,1)$ is fixed. We find exact conditions for holomorphicity in terms of the boundary behaviour of such functions. A by-product of our work is a new proof of the Berenstein–Pascuas two-radii theorem.
Keywords: holomorphicity, boundary behaviour.
Mots-clés : conformal automorphism 
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     author = {V. V. Volchkov and Vit. V. Volchkov},
     title = {Morera-type theorems in the hyperbolic disc},
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     pages = {31--60},
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     year = {2018},
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}
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V. V. Volchkov; Vit. V. Volchkov. Morera-type theorems in the hyperbolic disc. Izvestiya. Mathematics , Tome 82 (2018) no. 1, pp. 31-60. http://geodesic.mathdoc.fr/item/IM2_2018_82_1_a2/