Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Cooperation or Confrontation?

With globalization and increased connectedness, migration has become a political issue. Nevertheless, without citizenship participation in the political system in the host society is limited. Based on a neo-institutionalist approach and referring to political opportunity structure theory, this paper analyzes differences in means of political participation by migrants in two cities, one in the U.S. and one in Germany. Specific focus is put on the welfare state as one factor potentially influencing forms of participation of migrants. From interviews with migrants, local governments, and organizations, this paper establishes that political participation in the German city, Essen, is more institutionalized than in the U.S. city, Newark, NJ, where demonstrations and rallies play a more significant role. Looking at these findings, this paper explains the differences with a variation in the political opportunity structure between the two cities. Whereas in the conservative-corporatist welfare state, ideas of collective bar-gaining and the conferral of social rights to migrants leads to government-created bodies for migrant participation, in the U.S. city, these bodies do not exist and, therefore, migrants use different means of political participation. Through the conferral of social rights on migrants as well, in Germany, the cleavage between migrants and majority society has been pacified. In the U.S. city, where this is not the case, demonstrations and rallies are more common.