  We consider the inhomogeneous (or density dependent) incompressible Euler
equations in a three-dimensional periodic domain. We construct density
$\varrho$ and velocity $u$ such that, for any $\alpha<1/7$, both of them are
$\alpha $-H\"older continuous and $(\varrho, u)$ is a weak solution to the
underlying equations. The proof is based on typical convex integration
techniques using Mikado flows as building blocks. As a main novelty with
respect to the related literature, our result produces a H\"older continuous
density.
