  Let $\Delta$ be a (d-1)-dimensional simplicial complex and h^\Delta = (h_0^
,.. , h_d) its h-vector. For a face uniform subdivision operation \F we write
\Delta_\F for the subdivided complex and H_\F for the matrix such that
h^{\Delta_\F} = H_\F h^\Delta.
  In connection with the real rootedness of symmetric decompositions
Athanasiadis and Tzanaki studied for strictly positive h-vectors the
inequalities
  h_0 / h_1 \leq h_1 / h_{d-1} \leq .... \leq h_d / h_0 and h_1 / h_{d-1} \geq
... \geq h_{d-2} / h_2 \geq h_{d-1} / h_1.
  In this paper we show that if the inequalities holds for a simplicial complex
$\Delta$ and H_\F is TP_2 (all entries and two minors are non-negative) then
the inequalities hold for \Delta_\F.
  We prove that if \F is the barycentric subdivision then H_\F is TP_2. If \F
is the rth-edgewise subdivision then work of Diaconis and Fulman shows H_\F is
TP_2. Indeed in this case by work of Mao and Wang H_\F is even TP.
