Characteristic properties of Nehari functions

Abstract

Let N be the set of all meromorphic functions f defined in the unit disc D that satisfy Nehari's univalence criterion (1 − |z| 2 ) 2 |Sf (z)| ≤ 2. In this paper we investigate certain properties of the class N . We obtain sharp estimates for the spherical distortion, and also a two-point distortion theorem that actually characterizes the set N . Finally, we study some aspects of the boundary behavior of Nehari functions, and obtain results that indicate how such maps can fail to map D onto a quasidisc.

Schwarzian norms and two-point distortion

“…In the present paper we show more generally that for any δ > 0 the analytic functions with Schwarzian norm f ≤ 2(1 + δ 2 ) are characterized by the local distortion property (4) f (α, β) ≥ 1 δ sin(δ d(α, β)), α, β ∈ ‫,ބ‬ d(α, β) ≤ π δ .…”

“…The proofs follow an argument given by Chuaqui and Pommerenke [4] to show that the condition (3) implies f ≤ 2. It will suffice to carry out the details only for the condition (8), because the proof for (7) is quite similar.…”

“…Herold [13] had previously obtained a more general result for differential equations of higher order. The lower inequality is essentially contained in [3], and a proof is sketched in [4]. For completeness we include detailed proofs of both inequalities here.…”

“…Chuaqui and Pommerenke [4] gave a quantitative version of Nehari's theorem by showing that the condition f ≤ 2 implies that f has the two-point distortion property…”

Radius problems associated with pre-Schwarzian and Schwarzian derivatives

“…The last inequality leads to the implication S ⊂ B 1 (6). Note here that the Koebe function k(z) = z/(1 − z) 2 satisfies the relation (1 − z 2 )T k (z) = 2z + 4, which shows the inequality (1) is sharp.…”