Total positivity and two inequalities by Athanasiadis and Tzanaki

Let be a (d-1) -dimensional simplicial complex and h^ = (h₀^,. . , hd) its h-vector. For a face uniform subdivision operation we write _ for the subdivided complex and H_ for the matrix such that h^_ = H_ h^. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive h-vectors the inequalities h₀ / h₁ h₁ / h₃-₁. . . . hd / h₀ and h₁ / h₃-₁. . . h₃-₂ / h₂ h₃-₁ / h₁. In this paper we show that if the inequalities holds for a simplicial complex and H_ is TP₂ (all entries and two minors are non-negative) then the inequalities hold for _. We prove that if is the barycentric subdivision then H_ is TP₂. If is the rth-edgewise subdivision then work of Diaconis and Fulman shows H_ is TP₂. Indeed in this case by work of Mao and Wang H_ is even TP.