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

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 $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 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.