On some geometric properties of generalized Orlicz–Lorentz function spaces

In this article we continue investigations concerning generalized Orlicz–Lorentz function spaces ΛφΛφ initiated in the papers (Foralewski, 2011) [29] and [30] (cf. also (Foralewski, 2008) [27] and [28]). First, it is shown that modular ϱφϱφ is orthogonally subadditive. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space ΛφΛφ a strong condition of Δ2Δ2 type and the notion of regularity of the generated Musielak–Orlicz function φφ are introduced. Finally, criteria for non-squareness of ΛφΛφ are presented.