The geometric description of the holomorphic functions on the hyperbolic half-spaces (Q2901660)

In the given paper some useful generalizations of the so-called theorem of Dzjadyk about the geometrical description of holomorphic functions are discussed. It was well-known some conditions under which at least one of the functions \(u+iv,\) \(u-iv,\) defined in the domain \(\Omega\) of the complex plane \({\mathbb C}\) and taking the real values is holomorphic provided that \(u\) and \(v\) are continuously differentiable in \(\Omega\). Namely, the above conditions have the strictly relation to the equality between the squares of the surfaces of the graphics of the functions \(u,\) \(v\) and \(\sqrt{u^2+v^2}\). In the present paper, author gives some another conditions provided that \(\Omega\) is the hyperbolic analogue of the half-plane. The main result of the work is the following. For given \(\alpha\in [-\infty, \infty)\) we denote by \(H_{\alpha}\) some hyperbolic analogue of the half-space, and for given \(r_1\) and \(r_2>0,\) let \(\Phi_{r_i}(z),\) \(i=1,2,\) is fixed non-negative functions defined by some integral and having infinitely many zeros which to be denoted by \(N(r_1)\) and \(N(r_2),\) correspondingly. Let \(\varphi(u,v)\) be an arbitrary continuously differentiable function on \({\mathbb R}\) satisfying the condition \(\left(\frac{\partial\varphi}{\partial u}\right)^2+\left(\frac{\partial\varphi}{\partial v}\right)^2=1\). Suppose that \(N(r_1)\cap N(r_1)=\varnothing,\) the squares of the surfaces of the graphics of the each of the functions \(u,\) \(v\) and \(\varphi(u,v)\) equal to each other under all of the hyperbolic balls centered at arbitrary point \(z\) of the radii \(r_i\) in \(H_{\alpha}\) and, in addition, some three integral conditions hold. Then at least one of the functions \(u+iv,\) \(u-iv\) is holomorphic.