On the Uniformly Convexity

This paper discussed the characterizations of uniformly convexity of N -functions. Definition 1. A function M(u): R → R is called an N -function if it has the following properties: (1) M is even, continuous, convex; (2) M(0) = 0 and M(u) > 0 for all u = 0; (3) lim u→0 M(u) u = 0 and lim u→+∞ M(u) u = +∞. The N -function generates the Orlicz spaces. So it is important to analysis it. It is well-known that M(u) is an N -function iff M(u) = ∫ |u| 0 p(t)dt, where the right derivative p(t) of M(u) satisfies: (1) p(t) is right-continuous and nondecreasing; (2) p(t) > 0 whenever t > 0; (3) p(0) = 0 and lim t→∞ p(t) = +∞. Definition 2. A continuous function M : R→ R is called convex if M ( u+ v 2 ) 6 M(u) +M(v) 2 for all u, v ∈ R. If, in addition, the two sides of the above inequality are not equal for all u 6= v, then we call M strictly convex. Definition 3. For a continuous function M : R→ R. (1) If for any ε > 0 and any u0 > 0, there exists some δ > 0 such that M ( u+ v 2 ) 6 (1− δ) +M(v) 2 for all u, v ∈ R satisfying |u − v| > εmax{|u|, |v|} > εu0, then M is said to be uniformly convex for larger argument . 2000 Mathematics Subject Classification. Primary 46E30 Secondary 46B20.