# Minimum and maximum of the sum of a set raised to a power

Let's assume you have a set of S values, F1, F2, F3, ..., Fi, ... Fs such that you know the minimum and maximum possible values of Fi, but don't actually know any of the values Fi themselves. Let's call them $B_l$ for lower bound and $B_u$ for upper bound.

Now, assume I know the average of the set as well, $A = \frac{1}{S} \sum{F_i}$.

However, I'm interested in figuring out both the minimum and maximum of $\sum{F_i^q}$ as a function of A, Bl, Bu, and q.

For example, let's say I have 7 values. Their average is 6. These values I know fall between 0 and 20, but I don't actually know the values themselves, nor their actual min and max.

While I'm familiar with finding the min/max of a continuous function, I'm not totally where to even start with this one, and would appreciate any thoughts or suggested solution to the approach.

• do you want to find the min and max of the sum as a function of $q$? And what is $q$? Dec 21 '17 at 14:29
• if min(F) = m, max(F) = M, let X = max(|m|, |M|); then you can get some form of a bound involving X, q and S, but that would be a crude bound (although would be tight in certain settings)
– E-A
Dec 21 '17 at 14:31
• Sorry, I clarified that while the values are drawn from a range, I don't know the true min and max of this particular set. Dec 21 '17 at 14:32
• But the range is bounded? And do you have any knowledge on q at all?
– E-A
Dec 21 '17 at 14:33
• There's always Lagrange Multipliers. Dec 21 '17 at 14:37

Assuming $q \ge 1$ and non-negative $F_i$ i.e. $B_l \ge 0\,$, the generalized mean inequality gives:

$$A = \frac{1}{S}\sum F_i \;\le\; \sqrt[q]{\frac{1}{S}\sum {F_i}^q} \;\le\; \max \{F_i\} \;\le\; B_u \\[5px] \iff\quad A^q \;\le\; \frac{1}{S}\sum {F_i}^q \;\le\; B_u^{\,q}$$

You may amend the RHS inequality somewhat by noting that $\,\max\{F_i\}\le S \cdot A - (S-1)\cdot B_l\,$, but other than that there is not much room to improve, since the LHS inequality becomes an equality for $\,F_i = F_j\,$, and the RHS one for $\,F_i = B_u\,$, so both bounds can in fact be attained.