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Euclidean linear invariance and uniform local convexity

Part of: Entire and meromorphic functions, and related topics Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Wancang Ma  and
David Minda
Wancang Ma
Affiliation:
Department of Mathematical SciencesUniversity of CincinnatiCincinnati, Ohio 45221-0025, U.S.A.
David Minda
Affiliation:
Department of Mathematical SciencesUniversity of CincinnatiCincinnati, Ohio 45221-0025, U.S.A.
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Abstract

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Let S(p) be the family of holomorphic functions f defined on the unit disk D, normalized by f(0) = f1(0) – 1 = 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p. For p = ∞ these become the classical families S and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2p). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically k-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

MSC classification

Secondary: 30C99: None of the above, but in this section 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50: Coefficient problems for univalent and multivalent functions 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 3 , June 1992 , pp. 401 - 418
Copyright
Copyright © Australian Mathematical Society 1992

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