设x>0,y>0,a=x+y/1+x+y ,b=x/(1+x)+y/(1+y),则a,b的大小关系是?

设x>0,y>0,a=x+y/1+x+y ,b=x/(1+x)+y/(1+y),则a,b的大小关系是?
设x>0,y>0,a=x+y/1+x+y ,b=x/(1+x)+y/(1+y),则a,b的大小关系是?

设x>0,y>0,a=x+y/1+x+y ,b=x/(1+x)+y/(1+y),则a,b的大小关系是?
由于：
A
=(x+y)/(1+x+y)
=1-1/(1+x+y)
B
=x/(1+x)+y/(1+y)
=(x+2xy+y)/[(1+x)(1+y)]
=(x+2xy+y)/(1+x+y+xy)
=1-(1-xy)/(1+x+y+xy)
因为:
(1-xy)/(1+x+y+xy)
A

A=(x+y)/(1+x+y)=1-1/(1+x+y)
B=x/(1+x)+y/(1+y)=（x+2xy+y)/[(1+x)(1+y)
=（x+2xy+y)/(1+x+y+xy)=1-(1-xy)/(1+x+y+xy)
因为(1-xy)/(1+x+y+xy)<1/(1+x+y+xy)<1/(1+x+y)
所以-(1-xy)/(1+x+y+xy)>-1/(1+x+y)
故B>A

a-b
=(x+y/(1+x+y)-(x/(1+x)+y/(1+y))
=(x+y/(1+x+y)-(x+y+2xy)/(1+x+y+xy)
设x+y=m>0,1+x+y=n>0 xy=p>0
a-b
=m/n-(m+2p)/(n+p)
=[m(n+p)-n(m+2p)]/n(n+p)
=[mp-2np]/n(n+p)
=[(m-2n)p]/n(n+p)
<0
所以a