Sharp Distortion Theorems Associated with the Schwarzian Derivative

Chuaqui, Martin; Osgood, Brad

doi:10.1112/jlms/s2-48.2.289

Cited by 43 publications

(40 citation statements)

References 9 publications

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“…In fact, for no ε > 0 does the condition S f ≤ ε imply that f is a convex mapping. This can be seen by an example constructed in the paper [3]. There it is found that for 0 < t < 1 the function…”

Section: A Closer Look At Convex Mappingsmentioning

confidence: 82%

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Schwarzian derivatives of convex mappings

Chuaqui

Duren

Osgood

2011

Ann. Acad. Sci. Fenn. Math.

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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.

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Section: A Closer Look At Convex Mappingsmentioning

confidence: 82%

“…By a result of Chuaqui and Osgood [3], a normalized function f with S f ≤ 2 can not take the value −1/a 2 , so f 0 is analytic and univalent, with f 0 (0) = 0, f 0 (0) = 1, and f 0 (0) = 0. Also, S f 0 = S f , since the Schwarzian is preserved under Möbius transformation.…”

Section: Lemma 1 Let F Be a Convex Mapping Withmentioning

confidence: 99%

Schwarzian derivatives of convex mappings

Chuaqui

Duren

Osgood

2011

Ann. Acad. Sci. Fenn. Math.

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“…This leaves (1.1) invariant. The second normalization gives rise to the class N 0 , and according to a result in [3] if f satisfies (1.1) and (1.3) then it has no poles. In fact, such a function will either be a rotation of the logarithm…”

Section: Introductionmentioning

confidence: 99%

“…Much of the work in [3] is based on applying comparison theorems for solutions of differential equations to obtain bounds on f and f . For example, if f ∈ N 0 then u = (f ) −1/2 satisfies the initial conditions u(0) = 1, u (0) = 0, and it was shown that…”

Section: Introductionmentioning

confidence: 99%

Characteristic properties of Nehari functions

Chuaqui¹,

Pommerenke²

1999

Pacific J. Math.

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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.

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“…This class was studied in [2]. Obviously N ⊂ S. Related to N is the class N 0 defined as the family of functions in N such that f 00 (0) = 0.…”

Section: Introductionmentioning

confidence: 99%