Andreu Mas-Collel has also given two propositions in the direction of weakening the strong assumptions on the univalence condition of the Gale-Nikaido theorem.
j], that matters is that the determinant of the function be uniformly bounded away from zero in order to attain global univalence within the strictly positive orthant, [[R.
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.
Guney, On the univalence criterion of a general integral operator, J.
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.
Pescar, A new generalization of Ahlfors's and Becker's criterion of univalence, Bull.
12) gives a sufficient condition for close-to-convexity and hence for univalence
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 first result leads to sufficient conditions for univalence.
Nehari, Univalence conditions and Sturm-Liouville eigenvalues, Proc.