The planar Tur\'an number of $\{K_4,\Theta_5\}$

Let $\mathcal{F}$ be a set of graphs. The planar Tur\'an number, $ex_{\mathcal{P}}(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{F}$ as a subgraph. In this paper, we give upper bounds of $ex_{\mathcal{P}}(n,\{K_4,\Theta_5\})\leqslant25/11(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.