Convex functions and logarithms of polynomials with positive coefficients (Q1327542)

The author investigates functions \(f: \mathbb{R}^ d\to \mathbb{R}\cup \{- \infty, +\infty\}\) which can be obtained by an approximation process of the following type: there exists a sequence \(\{P_ n\}\) of generalized polynomials with positive coefficients such that \[ f(t_ 1,\dots, t_ d)= \lim_{n\to +\infty} {1\over n}\log P_ n(e^{t_ 1},\dots, e^{t_ d}) \] for each \((t_ 1,\dots, t_ d)\in \mathbb{R}^ d\). It is proved that any lower semi-continuous proper convex function \(f: \mathbb{R}^ d\to \mathbb{R}\cup \{+\infty\}\) has this property and that the convexity of a function \(f: \mathbb{R}\to \mathbb{R}\cup \{-\infty, +\infty\}\) is equivalent to this property. Furthermore, the author reveals relations between the conjugate function of \(f\) and the coefficients of the polynomials \(P_ n\) in the case when these coefficients satisfy certain regularity conditions. Finally, there are given theorems concerning the localization of the roots of the polynomials \(P_ n\) provided that these polynomials are the traces of the successive powers of a given matrix of polynomials.