  An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices
of $G$ into $b$-element subsets of some set of size $a$ in such a way that
$f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in
$G$. The fractional chromatic number $\chi_f(G)$ is the infimum of $a/b$ over
all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring.
Heckman and Thomas conjectured that the fractional chromatic number of every
triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami
and Zhu proved that $\chi_f(G) \leq 3-3/64 \approx 2.953$. Lu and Peng improved
the bound to $\chi_f(G) \leq 3-3/43 \approx 2.930$. Recently, Ferguson, Kaiser
and Kr\'{a}l' proved that $\chi_f(G) \leq 32/11 \approx 2.909$. In this paper,
we prove that $\chi_f(G) \leq 43/15 \approx 2.867$.
