In this paper we study the monotonicity and convexity properties in quasiBanach lattices. We establish relationship between uniform monotonicity, uniform C-convexity, H-and P L-convexity. We show that if the quasiBanach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly P L-convex; (ii) E is uniformly monotone; and (iii) E is uniformly C-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly C-convex of power type then it is uniformly Hconvex 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 P L-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly P L-convex if and only if both E and X are uniformly P L-convex.