  In this paper we present new embedding results for Musielak-Orlicz Sobolev
spaces of double phase type. Based on the continuous embedding of
$W^{1,\mathcal{H}}(\Omega)$ into $L^{\mathcal{H}_*}(\Omega)$, where
$\mathcal{H}_*$ is the Sobolev conjugate function of $\mathcal{H}$, we present
much stronger embeddings as known in the literature. Based on these results, we
consider generalized double phase problems involving such new type of growth
with Dirichlet and nonlinear boundary condition and prove appropriate
boundedness results of corresponding weak solutions based on the De Giorgi
iteration along with localization arguments.
