Abstract
Let G be the group of conformal automorphisms of the unit disc \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}\). For \(r>0\), we put \(B_{r}=\{z\in {\mathbb {D}}:|z|<\tanh r\}\). Denote by \(\overline{B}_{r}\) the closure of the disc \(B_r\), and by \(\partial B_{r}\) its boundary. Let \(\chi _r\) be the characteristic function (indicator) of \(B_r\). Assume that \(r_1, r_2\in (0,+\infty )\) are fixed and \(R>\max \, \{r_1,r_2\}\). We study the holomorphicity problem for functions \(f\in C(B_R)\) satisfying the condition
for all \(g\in G\) such that \(g \overline{B}_{r_j}\subset B_R\), \(j=1,2\). We find the exact conditions for holomorphicity in terms of size \(B_R\) and properties of zeros of generalized spherical transforms of functions \(\chi _{r_1}\) and \(\chi _{r_2}\). In particular, a strengthening of the Berenstein–Pascuas theorem (Israel J Math 86:61–106, 1994) on two radii is obtained.
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Dedicated to the memory of Professor Lawrence Zalcman (09.06.1943–31.05.2022).
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Volchkov, V., Volchkov, V. Zalcman’s problem and related two-radii theorems. Anal.Math.Phys. 13, 72 (2023). https://doi.org/10.1007/s13324-023-00835-5
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DOI: https://doi.org/10.1007/s13324-023-00835-5