Create some new numbers using $n$ arbitrary positive real numbers

Question

Known facts:

Let $a_i$, $b_i$, $i=1, \ldots, n$ be positive real numbers such that $a_1+ \cdots + a_n = a_1b_1 + \cdots +a_nb_n = 1$. Then $$b_1^{a_1}b_2^{a_2} \cdots b_n^{a_n} \leq 1.$$

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Let $c_i$, $i=1, \ldots, n$ be positive real numbers such that $c_1c_2 \cdots c_n = 1$. Then $$c_1^{c_1} c_2^{c_2} \cdots c_n^{c_n} \geq 1.$$

Questions: Let $x_1, x_2, \cdots, x_n$ be positive real numbers. Prove that $$x_1^{x_1} x_2^{x_2} \cdots x_n^{x_n} \leq \left( \frac{x_1^2+x_2^2+\cdots+x_n^2}{x_1+x_2+\cdots+x_n}\right)^{x_1+x_2+\cdots+x_n}.$$

It is a homework question. I would appreciate any help but not giving me the complete answers. Here comes my thoughts. In order to make use of the two facts, it is essential for me to construct the corresponding $a_i$, $b_i$ and $c_i$. This really bothers me. All I can think of is $$a_i = \frac{x_i}{x_1+\cdots+x_n}.$$ How should I proceed to find $b_i$ and $c_i$? Thanks in advance.

Answer 1

$S= \frac{x_1^2+x_2^2+\cdots+x_n^2}{x_1+x_2+\cdots+x_n}$

Try: $a_i = \frac{x_i}{x_1+\cdots+x_n}$ and $b_i = \frac{x_i}{S}$

Update

It works for sure, I just prove it.

Answer 2

Assign weights $x_i$ on $x_i$'s , i.e. the weight of $x_1$ is $x_1$; that of $x_2$ is $x_2$ and so on.

So then, the weighted AM of $(x_1, x_2, \dots, x_n)=\frac {{x_1}^2+{x_2}^2\dots+{x_n}^2}{x_1+x_2+\dots+x_n}$ and the weighted GM is ${({x _1}^{x_1}{x_2}^{x_2}\dots {x_n}^{x_n})^{\frac 1{x_1+x_2+\dots+x_n}}}$

As, weighted AM$\ge$weighted GM, hence, your claim follows.