  We prove that every $n$-vertex planar graph $G$ with no triangle sharing an
edge with a 4-cycle has independence ratio $n/\alpha(G) \leq 4 - \varepsilon$
for $\varepsilon = 1/30$. This result implies that the same bound holds for
4-cycle-free planar graphs and planar graphs with no adjacent triangles and no
triangle sharing an edge with a 5-cycle. For the latter case we strengthen the
bound to $\varepsilon = 2/9$.
