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Extra resources for Quasiconformal mappings and their applications. IWQCMA05
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If ρ(a) = 0, then v(a) = 0 < 1. Otherwise, ρ(a) > 0 and Kρ (a) ≤ −1. As a is a local maximum of v, it is also a local maximum of log v so that ∂ 2 log v (a) ≤ 0, ∂x2 ∂ 2 log v (a) ≤ 0. ∂y 2 We deduce that 0 ≥ △ log v (a) = △ log ρ (a) − △ log λr (a) = −Kρ (a)ρ(a)2 + Kλr (a)λr (a)2 ≥ ρ(a)2 − λr (a)2 . 2) This implies that v(a) ≤ 1, and completes the proof in the special case Ω = D. We now turn to the general case. Let h : D → Ω be a conformal mapping. Then h∗ (ρ(w)|dw|) := τ (z)|dz| is a C2 semimetric on D such that Kτ (z) ≤ −1 whenever τ (z) > 0.
3. Suppose ∆ and Ω are hyperbolic regions. 2) f ∆,Ω (w) + tanh d∆ (z, w) . 1 + f ∆,Ω (w) tanh d∆ (z, w) for all z, w ∈ ∆. Proof. 2) is trivial when f is a covering since both sides are identically one, Thus, it suffices to establish the inequality when f is not a covering of ∆ onto Ω. Then dD (0, f ∆,Ω (z)) ≤ dD (0, f ∆,Ω (w)) + dD (f ∆,Ω (z), f ∆,Ω (w)) ≤ dD (0, f ∆,Ω (w)) + 2d∆ (z, w) The hyperbolic metric and geometric function theory 47 gives 1 dD (0, f ∆,Ω (z)) 2 1 ≤ tanh dD (0, f ∆,Ω (w)) + d∆ (z, w) 2 f ∆,Ω (w) + tanh d∆ (z, w) = .
2 shows that the quasihyperbolic metric for a half-plane is the hyperbolic metric. 2. Suppose that Ω is a simply connected proper subregion of C. 1) λΩ (z) ≤ 2 , d(z, ∂Ω) and equality holds if and only if Ω is a disk with center z. Proof. Take any z0 in Ω, and let R = d(z0 , ∂Ω) and D = {z : |z − z0 | < R}. 1). If λΩ (z0 ) = 2/d(z0 , ∂Ω) then λΩ (z0 ) = λD (z0 ) so, by the Comparison Principle, Ω = D. The converse is trivial. 2 gives an upper bound on the hyperbolic metric of Ω in terms of the Euclidean quantity d(z, ∂Ω).