h****r
发帖数: 9 | 1 if f1(x)/g1(x) and f2(x)/g2(x) both decreasing, will (f1(x)+f2(x))/(g1(x)+g2
(x)) be also decreasing? If so, how to prove it? Thanks a lot. | H****h
发帖数: 1037 | 2 你是假设都是正函数吗？
g2
【在 h****r 的大作中提到】
: if f1(x)/g1(x) and f2(x)/g2(x) both decreasing, will (f1(x)+f2(x))/(g1(x)+g2
: (x)) be also decreasing? If so, how to prove it? Thanks a lot.
| h****r
发帖数: 9 | 3 yes. all functions are greater than 0.
【在 H****h 的大作中提到】
: 你是假设都是正函数吗？
:
: g2
| H****h
发帖数: 1037 | 4 应该不成立。令h=f/g。我们知道h1和h2是降的，但不知道
(h1g1+h2g2)/(g1+g2)是否降。注意到这个函数是h1和h2的凸组合。
在g1/g2很大的地方偏向于h1，在g2/g1很大的地方偏向于h2。
我们对g1和g2没有任何控制，也不知道h1和h2的大小关系。
【在 h****r 的大作中提到】
: yes. all functions are greater than 0.
| h****r
发帖数: 9 | 5 Well, it seems counter-intuition. | h****r
发帖数: 9 | 6 Well, it seems counter-intuition. | H****h
发帖数: 1037 | 7 很自然啊。
【在 h****r 的大作中提到】
: Well, it seems counter-intuition.
| h****r
发帖数: 9 | 8 Maybe it seems obvious to you. I don't see it.
Suppose f1,f2, g1,g2 all increasing. Then g1 increases faster than f1, g2
faster than f2. Wouldn't it be that g1+g2 increases faster than f1+f2?
(Sorry I can't input chinese on this computer). | z*****g
发帖数: 227 | 9 on (0,1) take positive functions
f1(x)=x(3-x),
f2(x)=(1-x)^2,
g1(x)=x,
g2(x)=1-x
f1/g1=3-x decreasing
f2/g2=1-x decreasing
(f1+f2)/(g1+g2)= x+1 increasing
g2
【在 h****r 的大作中提到】
: if f1(x)/g1(x) and f2(x)/g2(x) both decreasing, will (f1(x)+f2(x))/(g1(x)+g2
: (x)) be also decreasing? If so, how to prove it? Thanks a lot.
| h****r
发帖数: 9 | 10 Thanks.
【在 z*****g 的大作中提到】
: on (0,1) take positive functions
: f1(x)=x(3-x),
: f2(x)=(1-x)^2,
: g1(x)=x,
: g2(x)=1-x
: f1/g1=3-x decreasing
: f2/g2=1-x decreasing
: (f1+f2)/(g1+g2)= x+1 increasing
:
: g2
| x******i
发帖数: 3022 | 11
g2
suppose f1>>f2, but g2>>g1
then (f1+f2)/(g1+g2) ~ f1/g2
and there's no reason why f1/g2 decreases
for example f1=10000*x^3, g1=x^4, f2=x, g2=10000*x^2
then around x~1,
f1+f2 ~ 1000*x^3
g1+g2 ~ 1000*x^2
the ratio is ~ x, which increases with x.
【在 h****r 的大作中提到】
: if f1(x)/g1(x) and f2(x)/g2(x) both decreasing, will (f1(x)+f2(x))/(g1(x)+g2
: (x)) be also decreasing? If so, how to prove it? Thanks a lot.
| h****r
发帖数: 9 | 12 Good stuff. I got it.
【在 x******i 的大作中提到】
:
: g2
: suppose f1>>f2, but g2>>g1
: then (f1+f2)/(g1+g2) ~ f1/g2
: and there's no reason why f1/g2 decreases
: for example f1=10000*x^3, g1=x^4, f2=x, g2=10000*x^2
: then around x~1,
: f1+f2 ~ 1000*x^3
: g1+g2 ~ 1000*x^2
: the ratio is ~ x, which increases with x.
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