Convexity characteristic of Calderón–Lozanovskiĭ sequence spaces

Hudzik, Kamińska and Mastyło obtained some geometric properties of Calderón–Lozanovskiĭ function spaces which are defined on a nonatomic σ‐measure space ( Ω , Σ , μ ) in Houston. J. Math. 22 (1996), but left the case of atomic measure unsolved. We studied the relevant problems for the sequence spaces and obtained the following main results: For the Calderón–Lozanovskiĭ sequence spacese Φ , e Φis order continuous if and only ifΦ ∈ δ 2and e is order continuous . Let Φ be strictly convex on[ 0 , u b ] , then the convex characteristicε 0 ( e Φ ) = 2whenever e is not order continuous orΦ ∉ δ 2 ; if e is uniformly monotone andΦ ∈ δ 2 , thenε 0 ( e Φ ) ≤ 2 ( 1 − p ( Φ ) ) 1 + p ( Φ ). For the Orlicz‐Lorentz sequence spaceλ Φ , ω ,ε 0 ( λ Φ , ω ) = 2ifΦ ∉ δ 2orΨ ∉ δ 2 , or ω is not regular ;ε 0 ( λ Φ , ω ) = 2 ( 1 − p ( Φ ) ) 1 + p ( Φ )if Φ is strictly convex on0 , Φ − 11 ω 1, Φ ∈ δ 2and ω is regular .