Total positivity and two inequalities by Athanasiadis and Tzanaki

Let $\mathrm{\Delta }$ be a $(d-1)$-dimensional simplicial complex and $h^{\mathrm{\Delta }}=(h_0^{\mathrm{\Delta }},\dots ,h_d^{\mathrm{\Delta }})$ its $h$-vector. For a face uniform subdivision operation $\mathcal{F}$ we write ${\mathrm{\Delta }}_{\mathcal{F}}$ for the subdivided complex and $H_{\mathcal{F}}$ for the matrix such that $h^{{\mathrm{\Delta }}_{\mathcal{F}}}=H_{\mathcal{F}}{h}^{\mathrm{\Delta }}$. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive $h$-vectors the inequalities $\frac{h_0^{\mathrm{\Delta }}}{h_1^{\mathrm{\Delta }}}\le \frac{h_1^{\mathrm{\Delta }}}{h_{d-1}^{\mathrm{\Delta }}}\le \cdots \le \frac{h_d^{\mathrm{\Delta }}}{h_0^{\mathrm{\Delta }}}$ and $\frac{h_1^{\mathrm{\Delta }}}{h_{d-1}^{\mathrm{\Delta }}}\ge \cdots \ge \frac{h_{d-2}^{\mathrm{\Delta }}}{h_2^{\mathrm{\Delta }}}\ge \frac{h_{d-1}^{\mathrm{\Delta }}}{h_1^{\mathrm{\Delta }}}$. In this paper we show that if the inequalities holds for a simplicial complex $\mathrm{\Delta }$ and $H_{\mathcal{F}}$ is TP${}_2$ (all entries and two minors are non-negative) then the inequalities hold for ${\mathrm{\Delta }}_{\mathcal{F}}$. We prove that if $\mathcal{F}$ is the barycentric subdivision then $H_{\mathcal{F}}$ is TP${}_2$. If $\mathcal{F}$ is the $r$\textsuperscript{th}-edgewise subdivision then work of Diaconis and Fulman shows $H_{\mathcal{F}}$ is TP${}_2$. Indeed in this case by work of Mao and Wang $H_{\mathcal{F}}$ is even TP.