HMMT 2015 G5

Let $I$ be the set of points $(x, y)$ in the Cartesian plane such that

$$x>\left(\frac{y^{4}}{9}+2015\right)^{1 / 4}$$

Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^{2}+y^{2} \leq r^{2}$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<L r^{2}$ for all $r>0$.