Condenser Capacity and Meromorphic Functions

Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius

“…Then the inequalities (2.5) and (2.6) must be equalities. By [15] and the equality in (2.5), f must be univalent in D s . The equilibrium potential of the condenser (D s , K) is the function…”

“…Then the inequality (2.3) must be an equality. By [15], f must be univalent in D s and therefore Φ K = 0 on (0, s). Let k 0 = sup{s ∈ (0, 1) : there exists 0 < r < s such that Φ K (r) = Φ K (s)} > 0.…”

“…Then we must have equality in (3.2). By[15], f is univalent on sD and by (3.1) Φ R is constant and equal to | f (0)| on (0, s). Lets 0 = sup{s ∈ (0, 1) : there exists 0 < r < s such that Φ R (r) = Φ R (s)} > 0.Then Φ R is constant on (0, s 0 ) and strictly increasing on (s 0 , 1) and f is univalent on s 0 D.…”

Holomorphic Functions With Image of Given logarithmic or Elliptic Capacity

“…We denote by [0, z] the rectilinear segment with endpoints 0 and z. Since holomorphic functions reduce the capacity of condensers (see [13] and references therein),…”

“…Let 0 < r < s < 1. Since holomorphic functions reduce the capacity of condensers (see [13] and references therein), cap(sD, rD) ≥ cap( f (sD), f (rD)).…”