2] rows and columns are of the sign needed for the maximum suffices, I believe, to assure univalence of the equation set" (Ibid.
Gale, David and Hukukane Nikaido, "The Jacobian Matrix and Global Univalence of Mappings," Mathematische Annalen, 1965, 68-80, reprinted in Peter Newman, edited, Readings in Mathematical Economics, (Baltimore, MD: Johns Hopkins University Press, 1968), 81-93.
Bulut, Sufficient conditions for univalence of an integral operator defined by Al-Oboudi differential operator, J.
Pascu, On a univalence criterion (II), Itinerant seminar on functional equations, approximation and convexity, Cluj-Napoca, Preprint Nr.
There exist functions having infinitely many domains of univalence
(see for example ).
MITRINOVIC: On the univalence
of rational functions, Univ.
12) gives a sufficient condition for close-to-convexity and hence for univalence
A set G [subset] [DELTA] is called the set of local univalence
in the class A, if [[for all].
The problems of finding similar bounds on the Schwarzian derivatives that would imply univalence
, starlikeness or convexity of functions were investigated by a number of authors including Gabriel (4), Friedland and Nehari (3), and Ozaki and Nunokawa (11).
The proof is completed by applying the radius of univalence
for H and the properties of [F.
AKSENTIEV: Sufficient conditions for univalence
of regular functions.
Note that the result contained in Theorem 2 can be applied to obtain sufficient conditions for univalence
on the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity.