Morera-type theorems in the hyperbolic disc

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.