Abstract:
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.
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