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In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the  s -norms defined by the formula    $$\left\| x\right\| _{\Phi ,s}=\inf _{k&gt;0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$  x Φ , s = inf k &gt; 0 1 k s ∫ T Φ ( k x ) d μ , where  s  and    $$\Phi $$  Φ are an outer and Orlicz function respectively and  x  is a measurable real-valued function over a    $$\sigma $$  σ -finite measure space    $$(T,\Sigma ,\mu )$$  ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the  s -norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function  s  is strictly increasing or the measure    $$\mu $$  μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms