The minimum value of $ x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n}$ if $n$ is a positive integer

Question

For a fixed positive integer $n,$ compute the minimum value of the sum $$ x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n} $$ where $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are positive real numbers such that $$ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\ldots+\frac{1}{x_{n}}=n $$

I've been trying this problem for days but couldn't solve it.

The author of the book from this question is taken expects the reader to solve it via R.M.S.-A.M.-G.M.-H.M. inequalities and Weighted A.M.-G.M. inequalities.

One is free to answer it through advanced inequalities (Cauchy - Schwarz, etc.) but solving it through inequalities of means is more preferred.

I have done some hopeless tries which are not worthy of being included here.

What I have thought is that the inequalities of means preserve the degree in the RHS and the LHS of an inequality so maybe using only Means is not sufficient, maybe we can use the square root to find a minimum value.

Any help would be appreciated.,

Answer 1

Firstly, from AM-GM:

$$x_k+\frac{1}{x_k}\geq 2\Rightarrow x_k\geq 2-\frac{1}{x_k}$$

Using AM-GM one more time:

$$x_k^k+(k-1)=x_k^k+\underbrace{1+1+\ldots+1}_{k-1\ \text{times}}\geq k\sqrt[k]{x_k^k}=kx_k\geq k\left(2-\frac{1}{x_k}\right)$$

Thus:

$$\frac{x_k^k}{k}\geq 2-\frac{k-1}{k}-\frac{1}{x_k}=1+\frac{1}{k}-\frac{1}{x_k}$$

Summing, from $k=1$ to $n$:

$$\sum_{k=1}^n \frac{x_k^k}{k}\geq n+\sum_{k=1}^n\frac{1}{k}-\sum_{k=1}^n \frac{1}{x_k}=\sum_{k=1}^n\frac{1}{k}$$

Equality occurs when $x_k=1$.

Answer 2

In Application of weighted AM-GM it is shown that for non-negative real numbers $x_1, \ldots, x_n$ $$ \tag{*} x_1+\frac{x_2^2}{2}+\frac{x_3^3}{3}+\dots+\frac{x_n^n}{n}\ge H_n \cdot \left(x_1x_2x_3...x_n \right)^{1/H_n} $$ with $H_n = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$, the $n^\text{th}$ harmonic number.

In our case $$ 1 = \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\ldots+\frac{1}{x_{n}}} \le \sqrt[n]{x_1 x_2 \cdots x_n } $$ by the inequality between harmonic and geometric mean. It follows that $x_1 \cdots x_n \ge 1$, and substituting that in $(*)$ gives $$ x_1+\frac{x_2^2}{2}+\frac{x_3^3}{3}+\dots+\frac{x_n^n}{n}\ge1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n} \, . $$ Equality holds for $x_1 = \ldots = x_n = 1$.