Trying to find some properties of this set?

Question

I have a set of numbers, W_i, where

0
And I know that

sum( 1 / (1+W_i) ) = 1 (summed over all n, sorry but I can't remember LaTeX).


Define P_i with

P_i = W_i/4 .

From the information above, how do I find the boundaries of

sum(1/(1+P_i)) ???

Does it belong somewhere between 2 and 4?

Answer 1

sum(1/(1+P_i)) = 4 sum (1/(4+W_i))

So what we actually need is to find the range of
sum(1/(a_i+3)) subject to sum 1/a_i = 1 and a_i>1

For that I used the method http://en.wikipedia.org/wiki/Lagrange_multiplier of Lagrange multipliers
searching the extremum of
f(x1,...xn) = sum (1/(x_i+3)) subject to
sum 1/x_i =1
I got that there is a maximum when all x_i are equal with n,

that is, in original question, the maximum is
4n/(n+3)
I didn't get the minimum, probably is not attained in a fixed point though it might be one infimum( open interval)
Clearly 1/(4+W_i))>1/4(1+W_i), so one limit below is 1