By Vladimir N. Dubinin, Nikolai G. Kruzhilin

This is the 1st systematic presentation of the capacitory procedure and symmetrization within the context of complicated research. The content material of the e-book is unique – the most half has no longer been lined through current textbooks and monographs. After an advent to the idea of condenser capacities within the airplane, the monotonicity of the skill below a number of targeted adjustments (polarization, Gonchar transformation, averaging changes and others) is validated, by means of numerous sorts of symmetrization that are one of many major gadgets of the booklet. by utilizing symmetrization rules, a few metric homes of compact units are got and a few extremal decomposition difficulties are solved. furthermore, the classical and current evidence for univalent and multivalent meromorphic features are confirmed.

This e-book could be a invaluable resource for present and destiny researchers in a number of branches of advanced research and strength theory.

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Extra info for Condenser Capacities and Symmetrization in Geometric Function Theory

Example text

M and δk = 0 for k = m + 1, . . , n, where 1 m n − 1. 1, we set Γρ := (∂B(ρ)) \ ∂B. By the monotonicity of the capacity cap C(r; B, Γ, Z , for r ,Ψ ) cap C(r; B, Γ, Z, , Ψ) cap C(r; B(ρ), Γ ∪ Γρ , Z , ,Ψ ) ρ. 1. 40 Chapter 2. The Asymptotic Behaviour of the Capacity Asymptotic formulae for the capacity of a condenser at least two plates of which are not degenerating are consequences of the formulae above. This can be seen in the following example. 3. Let B be a ﬁnitely connected domain in C, {Ek }m k=1 be a system of pairwise disjoint closed sets Ek ⊂ B, k = 1, .

2 The case when one plate is nondegenerate We say that a parametric family of domains D(z0 , r), 0 < r < r0 , is asymptotically circular if the inclusions U (z0 , s(r)) ⊂ D(z0 , r) ⊂ U (z0 , t(r)) hold for some positive functions s(r) and t(r), 0 < r < r0 , such that s(r) ∼ t(r) ∼ r as r → 0. An element D(z0 , r) of an asymptotically circular family is called an almost disc with centre z0 and radius r, and we shall call the intersection of an almost disc with a half-plane whose boundary contains z0 an almost half-disc.

N, are positive numbers. 15) 2 (δk − δ) + R(C, ∅, Z, νk log r where , Ψ) n δ= 1 log r +o n δk /νk k=1 2 1/νk , k=1 1 log r 2 , r → 0, 46 Chapter 2. The Asymptotic Behaviour of the Capacity and R(C, ∅, Z, , Ψ) n = 2π k=1 (δk − δ)2 log μk + νk2 n n k=1 l=1 l=k (δk − δ)(δl − δ) log |zk − zl | . νk νl A prime on a summation sign means that if zk = ∞ (zl = ∞), then the corresponding term is set equal to zero. Proof. We can assume that δ = 0 and δn = 0. Now if zk = ∞, k = 1, . . 4 replacing the functions nB (z, zk , 0) with nC (z, zk , 0) = log |zzk /(z − zk )|, k = 1, .