Applications of special function theory to complex analysis

For any two points a₁ and a₂ in an open disk Δ on the complex sphere C¯, let L be a curve separating a ₁ from a₂ on C¯, which splits C¯ into two complementary regions B ₁ ∋ a₁ and B₂ ∋ a₂. Let l be a part of this curve lying on . In this note we study the question of how small can the average harmonic measure (1/2)(ω(a₁, l, B₁) + ω(a₂, l, B₂)) be? Here ω(a _k, l, B_k) denotes the harmonic measure of l with respect to B_k at the point a_k. This question can be interpreted as a problem on the minimal average temperature at two points in a long cylinder, composed by two media, separated by a heating membrane, each of which contains a reference point.