The mathematical relation between the impact factor and the uncitedness factor

In a general framework, given a set of articles and their received citations (time periods of publication or citation are not important here) one can define the impact factor (IF) as the total number of received citations divided by the total number of publications (articles). The uncitedness factor (UF) is defined as the fraction of the articles that received no citations. It is intuitively clear that IF should be a decreasing function of UF. This is confirmed by the results in [VAN LEEUWEN & MOED, 2005] but all the given examples show a typical shape, seldom seen in informetrics: a horizontal S-shape (first convex then concave). Adopting a simple model for the publication-citation relation, we prove this horizontal S-shape in this paper, showing that such a general functional relationship can be generally explained.