An improved upper bound for the planar Turán number of $C_8$

We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most $$\frac{69}{25}(n-2)$$ edges, for every $n\ge 8$. This improves the best known bound $$\frac{323}{108}n-6\text{for every }n\ge 27.$$