关于N-函数的指标函数的单调性定理

On the montionial theorems of index function of Nfunctions

研究生成N-函数的两类数量指标的指标函数的单调性及其关系 ,主要结果为 :(1)设Φ为N -函数 ,φ(t)为它的左导数 ,记 FΦ(t) =tφ(t)Φ(t) , GΦ(c ,u) =Φ- 1(u)Φ- 1(cu) (c>1) ,则FΦ(t)在 (0 .Φ- 1(u0 ) ]单调递增当且仅当对任意c >1,GΦ(c,u)在 (0 ,u0c]单调递增 .(2 )设Φ ,Ψ为一对互余的N -函数 , ,ψ分别为它们的左导数 ,则 (i)FΦ(t)在 (0 ,ψ(C) ]上单调递增 (递减 )当且仅当FΨ(s)在 (0 ,C]上单调递减 (递增 ) ;(ii) 1/a Φ + 1/b Ψ =1.

We studied the relationship of the couple of index function of Nfuctions,The main results are:(1)Let F Φ(t)=tφ(t)Φ(t). G Φ(c,u)=Φ -1 (u)Φ -1 (cu)(c>1), where Φ is an Nfunction and φ(t) is its left derivative,Then F Φ(t) is increasing on (0,Φ -1 (u 0)] iff G Φ(c,u) is increasing on (0,u 0c] for all c>1. (2)Let Φ,Ψ be a pair of complementary Nfunctions,with φ,ψ are their left derivatives.Then (i) F Φ(t) is increasing (decreasing) on (0,ψ(c)] iff F Ψ(s) is decreasing (increasing) on (0,c].(ii)1a * Φ+1b * Ψ=1,where a * Φ=inf｛F Φ(t):t∈(0,Φ(c)]｝,b * Ψ=sup｛F Ψ(s):s∈(0,c]｝.