= Coefficient-dominated entrywise positivity
Let real <power series> $f(t)=\sum_{k\geq0}f_kt^k$ and $g(t)=\sum_{k\geq0}g_kt^k$ converge on $[-1,1]$, with $f_k\geq|g_k|$ for every index including zero. If $X\succeq0$ has entries there, applying $f$ within two diagonal blocks and $g$ across them preserves <positive semidefiniteness>. Indeed the <coefficient> <matrix> $H_k$ is a <bipartite block-constant positive semidefinite matrix>. Each $H_k\circ X^{\circ k}$ is <positive semidefinite> by the <Schur product theorem>, and their sum converges to the transformed <matrix>. Closedness of the <positive semidefinite cone> completes the proof. Convergence at one gives $\sum f_k<\infty$, so <coefficient> domination ensures <absolute convergence> of both series throughout the interval. Without the zero-index condition, $f=-1$, $g=0$ and $X=0$ give a counterexample.
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