Abstract
Let Σ(Ω) be the class of functions [Formula presented] univalent on a finitely connected domain Ω, ∞∈Ω⊂C‾. By a classical result due to H. Grötzsch, the function f0 maximizing ℜa1 over the class Σ(Ω) maps Ω onto C‾ slit along horizontal segments. Recently, M. Bonk found a similar extremal problem, which maximizer f1∈Σ(Ω) maps Ω onto a domain on C‾, whose complementary components are squares. In this note, we discuss a parametric family of extremal problems on the class Σ(Ω) with maximizers fm, 0<m<1, mapping Ω onto domains on C‾, whose complementary components are rectangles with horizontal and vertical sides and with module m.
Original language | English |
---|---|
Article number | 123927 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 486 |
Issue number | 2 |
DOIs | |
State | Published - Jun 15 2020 |
Keywords
- Canonical domain
- Conformal mapping
- Extremal problem
- Rectangle
- Uniformization