Generalization and application of certain Radon's theorem (Q677725)

Let \(L\) be a closed curve represented in the form \(z(s)=z_0+ \int_{0}^{s}e^{i\theta (t)}dt\), \(|s|\leq s_0\leq \infty\), \(\theta\) is a real function continuous from the left. Set \(\alpha (L)=\int_{-s_0}^{s_0}|d\theta (t)|\). For \(w\notin L\) let \(\omega_w(t)\) be a continuous branch of \(\arg (z(t)-w)\) and put \(V_w(L)=\int_{-s_0}^{s_0}|d\omega_w(t)|\). The curve \(L\) is called Radon curve if \(\alpha (L)<\infty\) [see, \textit{J. Radon} [Sitzungsberichte, Akad. Wiss., Wien 128, 1123-1167 (1920; JFM 47.0457.01)]. The author introduces generalized Radon curves as a combination of the Radon curves passing through infinity. Let \(t_1,\dots, t_n\) be all the points where \(z(t_k)=\infty\). Every curve may be parametrized with respect to its natural parameters. Theorem 1 of the article states the inequality \(V_w(L)\leq \alpha (L)+\pi n\) for a generalized Radon curve \(L\). The equality is attained if and only if the corresponding functions \(\theta _k\) and \(\omega_{w}^{k}\) of the combinated Radon curves are equimonotone. Let \(D_m\) be a domain bounded by \(m\) curves \(l_j\) having right and left tangents at any of its point. \(f\in R(D_m)\) if \(f\) is meromorphic in \(D_m\), generalized continuous in its closure and \(f(l_j)\) its Radon curve for any \(j\). Theorem 2 of the article estimates a difference between the numbers of \(w\)-points and \(\infty\)-points of \(f\in R(D_m)\). There are some geometrical additions.