The theorem of Fellman and Jakobsson: a new proof and dual theory

Abstract

The Fellman and Jakobsson theorem of 1976 deals with transformations phi of the rank-frequency function g and with their Lorenz curves L (phi degrees g) and L(g). It states (briefly) that L (phi degrees g) is monotonous (in terms of the Lorenz dominance order) with phi(chi)/chi. In this paper we present a new, elementary proof of this important result. The main part of the paper is devoted to the dual transformation g degrees psi(-1), where psi is a transformation acting on source densities (instead of item densities as is the case with the transformation phi ). We prove that, if the average number of items per source is changed after application of the transformation psi, we always have that L(g degrees psi) and L(g) intersect in an interior point of[0, 1], i.e. the theorem of Fellman and Jakobsson is not true for the dual transformation. We also show that this includes all convex and concave transformations. We also show that all linear transformations psi yield the same Lorenz curve. We also indicate the importance of both transformations phi and psi in informetrics.