An improved upper bound for the planar Tur\'an 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 \[
\frac{323}{108}n-6
\qquad \text{for every } n\ge 27. \]