The maximum number of two 4-vertex graphs in planar graphs

Let $f(n,H)$ be the maximum number of copies of a graph $H$ in a planar graph of order $n$. When $H$ is a connected graph on four vertices, $f(n,H)$ has been completely determined except for two cases: $K_{1,3}^{+}$ (the claw graph $K_{1,3}$ with one additional edge) and $K_4^{-}$ (the complete graph $K_4$ with one edge removed). Here, we address these two cases and establish that for all $n\ge 4$, $$f(n,K_{1,3}^{+})=4n^2-12n-4\text{ and }f(n,K_4^{-})=\frac{1}{2}(n^2+9n-40)\mathrm{.}$$