Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform C-convexity, H- and PL-convexity. We show that if the quasi-Banach lattice E has a-convexity constant one for some 0 < a < C, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly C-convex. In particular, it is shown that if E has a-convexity constant one for some 0 < a < c and if E is uniformly C-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved. Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Kothe function space with a-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Kothe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.