送交者: Yush 于 2005-7-26, 01:53:59:
回答: A better proof using only high school knowledge 由 Yush 于 2005-7-26, 01:14:39:
I think the convexity of the logarithm could mean the inequality we learned in high school:
(0) log(x) <= x-1, where the equality holds iif x == 1
which means the logarithm curve y = log(x) is always above the line y = x-1, with only one intersection at x = 1.
Some extra conditions must be given, like Xi and Yi must be positive. Yi should be positive so that its logarithm exists; so should Xi, so that the second partial derivative of Z with respect to Yi, (=-Xi/Yi^2), could be negtive to ensure the existance of the maximum of Z. Now, let's assume these.
(1) Z - Sum( Xi*log(Xi*Y/X) )
(2) = Sum( Xi*log(Yi) ) - Sum( Xi*log(Xi*Y/X) )
(3) = Sum( Xi*log( X*Yi/(Xi*Y) ) )
(4) <=Sum( Xi*( X*Yi/(Xi*Y) - 1 ) )
(5) = X/Y*Sum(Yi) - Sum(Xi)
(6) = 0
In step (4), we use inequality (0). Note, Xi's have to be positive for this step. The equality of this step holds iif X*Yi/(Xi*Y) = 1, as what inequality (0) states.
This actually completes the proof. You may need further explaination. From the above proof, we have
Z - Sum( Xi*log(Xi*Y/X) ) <= 0, where the equality holds iif X*Yi/(Xi*Y) = 1
In other words, Z reaches its its maximum, Sum( Xi*log(Xi*Y/X) ), iif Yi = (Y/X) * Xi.