Let $x_1, x_2, ... ,x_n$ be positive real numbers and define $S(k)$ to be the power sum $S(k) = x_1^k + x_2^k +... + x_n^k$ . It is given that $S(3) = 3$ and that $S(5) = 5 $. Find the best lower bound for $S(1) $.


(1) An application of Cauchy Schwarz shows that $S(1) > 9/5$ is an admissible lower bound.

(2) The problem appears (with a smaller bound) in the book "Inequalities" by Cvetkovski (p. 360).



It seems the following.

Since the domain of admissible values of $x$ is compact, the sum $S(1)$ attains its minimum. To find it we apply the method of Lagrangian multiplies. Put $$\mathcal L=S(1)+\lambda(S(3)-3)+\mu(S(5)-5).$$

Then $$\frac{\partial{\mathcal L}}{\partial{x_i}}=1+3\lambda x_i^2+5\mu x_i^4.$$

Hence the minimum is attained when some of $x_i$ are at the relative boundary of the domain (that is equals zero), and the others are roots of the equation $1+3\lambda x^2+5\mu x^4$. Hence $k$ of $x_i$ equals $a>0$ and $l$ of $x_i$ equals $b>0$. Then $ka^3+lb^3=3$ and $ka^5+lb^5=5$. Solving this system numerically (by Mathcad), suggests that the minimum value of $S(1)=2.1367\dots$ is attained when $k=l=1$ and $a=1.3642\dots.$


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