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

The Fellman and Jakobsson theorem of 1976 deals with transformations φφ of the rank–frequency function gg and with their Lorenz curves L(φ∘g)L(φ∘g) and L(g)L(g). It states (briefly) that L(φ∘g)L(φ∘g) is monotonous (in terms of the Lorenz dominance order) with φ(x)x. 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∘ψ−1g∘ψ−1, where ψψ is a transformation acting on source densities (instead of item densities as is the case with the transformation φφ). We prove that, if the average number of items per source is changed after application of the transformation ψψ, we always have that L(g∘ψ)L(g∘ψ) and L(g)L(g) intersect in an interior point of [0,1][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 ψψ yield the same Lorenz curve.We also indicate the importance of both transformations φφ and ψψ in informetrics.