Harmonic pairs of functions on subsets of the complex plane (Q2901597)

The article presents a generalization of a theorem due to V. K. Dzjadyk on geometrical description of holomorphic functions:NEWLINENEWLINE Let \(S\left(f,r,z\right)\) be the area of the surface of the graph of a function \(f\in C^1\) lying over the ball centered at \(z\) of the radius \(r\). Denote \(f_1=u\), \(f_2=v\), \(f_3=\sqrt{u^2+v^2}\), \(z=x+iy\), \(x=\mathrm{ Re} z\), \(y=\mathrm{ Im} z\), \(\Lambda=\left\{\frac{\alpha}{\beta}: J_{1}(\alpha)=J_{1}(\beta)=0,\alpha,\beta>0\right\}\), where \(J_{\nu}\) denotes a Bessel function of the first type of order \(\nu\) and \(i,j,k\in\{1,2,3\}\). Suppose that \(u,v\in C^1(\mathbb C)\), \(r_1\) and \(r_2\) are fixed positive numbers with \(\frac{r_1}{r_2}\notin\Lambda\) and \(S\left(f_i,r_1,z\right)=S\left(f_j,r_1,z\right)\), \(S\left(f_j,r_2,z\right)=S\left(f_k,r_2,z\right)\) for all \(z\in {\mathbb C}\). Suppose that there exists some sequence \(M_q\) of positive numbers with \(\sum\limits_{m=1}^{\infty} \left(\inf\limits_{q\geq m}M_q^{1/q}\right)=+\infty\) and \(\gamma>0\) with \(\int\limits_{-\infty}^{\infty}\left| S\left(f_j,r_1,z\right)-S\left(f_k,r_1,z\right)\right| \left(1+| x| ^q\right)dx\leq M_q\cdot\exp{\gamma| y| }\) and \(\int\limits_{-\infty}^{\infty}\left| S\left(f_i,r_2,z\right)-S\left(f_j,r_2,z\right)\right| \left(1+| x| ^q\right)dx\leq M_q\cdot\exp{\gamma| y| }\) for every \(q\in {\mathbb N}\) and \(y\in {\mathbb R}\). Then at least one of the functions \(u+iv\) or \(u-iv\) is analytic in \({\mathbb C}\).NEWLINENEWLINE Some other results are formulated in the paper, too. In particular, the corresponding case of a half-plane is also considered.