Boundary distortion and variation of the module under an extension of a doubly connected domain

Theorem 2

Let f(z) ∈ $\mathcal{F}(\rho ,r)$ , f(z) ≠ e^iα f(z;pr), α ∈ ?, and let ?(t) be a strictly convex monotone function of t>0. Then $$\int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } )|)d\theta< } \int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } ;\rho ,r)|)d\theta } $$ . The proof of this theorem is based on the Golusin-Komatu equation. If E is a continuum in the disk U_R={z:|z|<R}, then M (R, E) denotes the conformal module of the doubly connected component of U_R/E; let $\varepsilon (m) = \{ E:\overline U _r \subset E \subset U_1 , M(1,E) = M^{ - 1} \} $ .

Problem 3

Find the maximum of M(R, E), R>1, and the minimum of cap E over all E in ε(m). This problem was posed by V. V. Kozevnikov in a lecture to the Seminar on Geometric Function Theory at the Kuban University in 1980, and by D. Gaier (see 2]). The solution of this problem is given by the following theorem.

Theorem 3

Let $E^* = \underline U _m \cup m,s]$ . If R>1; E, E^* ∈ ε(m) and E ≠ e^iα E*, α ∈ ?, then M(R, E)<M(R, E^*), capE^*<capE. A similar statement is also proved for continua lying in the half-plane. Bibliography: 7 titles.