Some monotonicity properties in F-normed Musielak–Orlicz spaces

Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces $$L^\Phi (\mu )$$ L Φ ( μ ) endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces $$E^\Phi (\mu )$$ E Φ ( μ ) with the F-norm induced from $$L^\Phi (\mu )$$ L Φ ( μ ) . The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.