Convex order for convolution polynomials of Borel measures (Q2315106)

If for two finite Borel measures \(\mu\), \(\nu\) with finite first moment the following inequality \[ \int\limits_{\mathbb R} \varphi(x) \mu(dx) \leq \int\limits_{\mathbb R} \varphi(x) \nu(dx) \] holds for any convex function \(\varphi : {\mathbb R} \rightarrow {\mathbb R}\), then it is said that \(\mu\) is smaller than \(\nu\) in the convex order (denoting \(\mu \leq_{cx} \nu\)). Analogously, if two real polynomials \(P\) and \(Q\) of \(m\) variables are considered as convolution polynomials with respect to finite Borel measures \(\mu_1, \ldots, \mu_m\), then the respective inequality leads to the notation \(P(\mu_1,\ldots,\mu_m) \leq_{cx} Q(\mu_1,\ldots,\mu_m)\) (i.e., to the notion of the convex order inequality for the polynomial distribution). The conditions on polynomials with nonnegative coefficients \(P(\mu_1,\ldots,\mu_m) \leq_{cx} Q(\mu_1,\ldots,\mu_m)\) are found to hold . Some open problems posed by I. Rasa are solved and new open questions are formulated.