Uniform Gateaux differentiability and weak uniform rotundity in Musielak–Orlicz function spaces of Bochner type equipped with the Luxemburg norm

Criteria for weak uniform rotundity and reflexivity of Musielak–Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. Although, criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak–Orlicz function spaces of real functions equipped with the Luxemburg norm were known, they can be also easily deduced from our main results.