Natural exponential families associated to Pick functions

We define a quadratic action of the group of invertible (2,2) matrices of determinant 1 by Mœbius transformsh(x)=(ax+b)/(cx+d) on the natural exponential families (NEF) on ℝ which changes the mean functionk′ of the NEFF in the new mean functionh(k′) associated to the new NEF, denoted byh(F). The variance function ofh(F) is(cm+d)2VF(h(m)). Whenz→k′(z) orz→a k′ (a logz) happens to be a Pick function,h(F) can be explicitely described. We prove that certain cubic NEF belong to this type. This fact leads us to a classification of the variance functionsP(m)/m, where the polynomialP has degree ≤3 without complex zeros.