  We give here some new lower bounds on the order of a largest induced forest
in planar graphs with girth $4$ and $5$. In particular we prove that a
triangle-free planar graph of order $n$ admits an induced forest of order at
least $\frac{6n+7}{11}$ , improving the lower bound of Salavatipour [M. R.
Salavatipour, Large induced forests in triangle-free planar graphs, Graphs and
Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order
$n$ and girth at least $5$ admits an induced forest of order at least
$\frac{44n+50}{69}$.
