A lower bound on the order of the largest induced linear forest in triangle-free planar graphs

We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n-2m}{11}$ vertices, and thus at least $\frac{5n+8}{11}$ vertices. Furthermore, we show that there are triangle-free planar graphs on $n$ vertices whose largest induced linear forest has order $\lceil \frac{n}{2}\rceil +1$.