Bloch estimates in non-doubling generalized Orlicz spaces

We study minimizers of non-autonomous functionals

$$\begin{array}{r}\underset{u}{inf}{\int }_{\mathrm{\Omega }}\phi (x,|\mathrm{\nabla }u|)dx\end{array}$$

when $\phi$ has generalized Orlicz growth. We consider the case where the upper growth rate of $\phi$ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $\phi$ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.