  P\'olya's Positivstellensatz on the $1$-simplex says that if $P(x)$ is a real
polynomial such that $P(x)>0$ whenever $x \ge 0$, then all the coefficients of
$(1+x)^mP(x)$ are positive whenever $m$ is large. Powers-Reznick gave a
complexity estimate for P\'olya's Positivstellensatz. Namely, they proved that,
for such $P(x)$ of degree $d$, all the coefficients of $(1+x)^mP(x)$ are
positive whenever $m > \frac{1}{2} (d^2 -d) \frac{L(P)}{\lambda(P)} - d$. where
$\frac{L(P)}{\lambda(P)}$ is an invariant of $P(x)$. For $d=3$ and $d=4$
specifically, we improve Powers-Reznick's bound by showing $m > \frac{3}{2}
\frac{L(P)}{\lambda(P)} - 1$ for $d=3$ and $ m > \frac{4232}{2505}
\frac{L(P)}{\lambda(P)} - 1$ for $d=4$.
