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

Let F(p, τ) denote the class of univalent analytic functions f(z) in the domain K(p, 1) = {z : p < |z| < 1}, satisfying |f(z)| = 1 for |z| = 1 and τ |f(z)| < 1 for z ∈ K(p, 1). Let f(z;p,r) map K(p, 1) onto the domain K(p, 1) \ [τ, s] and let f(z; p, r) ∈ F (p, r). Theorem 2. Let f(z) ∈ F(p,r), f(z) ≠ e^ia f(z;pτ), α ∈ ℝ , and let Φ(t) be a strictly convex monotone function oft > 0. Then (Equation Presented) 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 ε(m) = {E:Ū τ ⊂ E ⊂ U₁, M(1,E) = M^-1}. Problem. Find the maximum of M(A, 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^* = U_m U [m, s]. If R > 1; E,E* ε ε(m) and E ≠ e^ia E^*, α ∈ ℝ, 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.