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