Schwarzian derivatives of convex mappings

Abstract

Abstract. A simple proof is given for Nehari's theorem that an analytic function f which maps the unit disk onto a convex region has Schwarzian norm S f ≤ 2. The inequality in sharper form leads to the conclusion that no convex mapping with S f = 2 can map onto a quasidisk. In particular, every bounded convex mapping has Schwarzian norm S f < 2. The analysis involves a structural formula for the pre-Schwarzian of a convex mapping, which is studied in further detail.

Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings

“…This was also derived in [2] as a consequence of the invariant form of Schwarz's lemma. Equality occurs in (9) at a point z 0 = 0 if and only if ϕ is a unimodular constant, in which case f is a slit mapping, or if ϕ is a Möbius transformation of D onto itself.…”

“…In [2] we proved an analog of Theorem 2 for mappings onto the interior of convex polygons, though we did not explicitly note the connection with pre-vertices. We state the result as follows: there is no normalization at the origin, where the function is analytic, and the polygon can be bounded or unbounded.…”

“…We state the result as follows: there is no normalization at the origin, where the function is analytic, and the polygon can be bounded or unbounded. The proof is based on the fact, established in [2], that such mappings are characterized by the representation…”

A note on convex conformal mappings

“…see [6] and [3]. The results in [2] thus apply, namely that λ has at most one critical point, with the exception of a parallel strip where ∇λ = 0 all along the central line.…”

On a theorem of Haimo regarding concave mappings

“…This result may be compared with another theorem of Nehari [13] (see also [3]) that every convex mapping has the Schwarzian norm Sf ≤ 2. For the full class of concave mappings (3), however, nothing better than Sf ≤ 6 is true, since the mapping onto the complement of a linear segment is concave and has Sf = 6.…”