Kuznetsov, V. O. Properties of the conformal radius of a domain. (English. Russian original) Zbl 1130.30301 J. Math. Sci., New York 118, No. 1, 4871-4879 (2003); translation from Zap. Nauchn. Semin. POMI 276, 237-252 (2001). Summary: We consider the function \(\rho (z) = R(D,z)\), where \(R(D,z)\) is the conformal radius of a simply connected domain \(D\) at a point \(z \in D\). We study relations between the values of the function \(\rho (z)\) at various points of the domain \(D\). In Sec. 1, we establish exact inequalities relating the values of the function \(\rho (z)\) in an arbitrary simply connected domain \(D \subset \overline {\mathbb{C}}\) with the position of the conformal center and with the maximal conformal radius of the domain \(D\). The same values are related to the values of \(\rho (z)\) at another two points of the domain \(D\). In Sec. 2, similar results are established for convex domains. This work supplements some recent results of E. G. Emel’yanov [J. Math. Sci., New York 89, No. 1, 976–987 (1998); translation from Zap. Nauchn. Semin. POMI 226, 93–108 (1996; Zbl 0907.30025)], L. V. Kovalev [J. Math. Sci., New York 110, No. 6, 3111–3120 (2002); translation from Zap. Nauchn. Semin. POMI 263, 141–156 (2000; Zbl 1002.30014)] and H. R. Haegi [Compos. Math. 8, 81–111 (1950; Zbl 0041.05101)]. MSC: 30C35 General theory of conformal mappings 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods Citations:Zbl 0907.30025; Zbl 1002.30014; Zbl 0041.05101 × Cite Format Result Cite Review PDF Full Text: DOI