Monotonicities in Orlicz Spaces Equipped with Mazur-Orlicz F-Norm

Abstract

Some basic properties in Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz F-norm are studied in this paper. We give some relationships between the modulus and the Mazur-Orlicz F-norm. We obtain an interesting result that the norm of an element in line segments is formed by two elements on the unit sphere less than or equal to 1 if and only if that the monotone function is a convex function. The criterion that Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz F-norm are strictly monotone or lower locally uniform monotone is presented.

Lower Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

“…It is worth noting that quasi-Banach spaces have been extensively studied over the last century (see [1][2][3][4][5][6]). As we know, in the realm of quasi-Banach spaces, the geometry is heavily influenced by the significant role played by monotonicity properties.…”

“…≥ k 2 . In virtue of the inequality(2), there is a positive value d that satisfies the inequalityΦ(t 0 , x m k (t 0 ) ∥x m k ∥ F ) + d ≤ Φ(t 0 , x(t 0 ) ∥x m k ∥ F ) holds.The above inequality contradicts with Equality (1). So, in this case, we havelim k→∞ x m k (t 0 ) ∥x m k ∥ F = x(t 0 ) ∥x∥ F .Case 2.…”

Uniformly Nonsquare in Orlicz Space Equipped with the Mazur-Orlicz F -Norm

“…In 2018, Cui et al discussed the monotonicity of Orlicz space that generated by the monotone continuous function equipped with Mazur-Orlicz F-norm (see [1,2]). In 2020, Bai et al given criteria that Orlicz spaces that generated by the monotone function equipped with Mazur-Orlicz F-norm have strictly monotonicity and upper locally uniform monotonicity, and they get the conclusion that kλx + ð1 − λÞyk F ≤ 1 for each x, y ∈ SðL Φ ðμÞÞ and λ ∈ ð0, 1Þ if and only if Φ is convex function on R. So, in order to studying geometric properties of Orlicz spaces equipped with the Mazur-Orlicz F-norm, we need to assume that Φ is convex see [3].…”