Problem 1656

Proof: Let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers (to be determined). According to the Cauchy-Schwarz inequality, we have
$$\left(\frac{x_{1}^{2}}{a_{1}}+\frac{x_{2}^{2}}{a_{2}}+\cdots+\frac{x_{k}^{2}}{a_{k}}\right)\left(a_{1}+a_{2}+\cdots+a_{k}\right) \geq\left(x_{1}+x_{2}+\cdots+x_{k}\right)^{2}$$

The inequality can be rewritten as
$$\left(\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}\right)^{2} \leq \frac{a_{1}+a_{2}+\cdots+a_{k}}{k^{2} a_{1}} x_{1}^{2}+\frac{a_{1}+a_{2}+\cdots+a_{k}}{k^{2} a_{2}} x_{2}^{2}+\cdots+\frac{a_{1}+a_{2}+\cdots+a_{k}}{k^{2} a_{k}} x_{k}^{2}$$

For $k=1,2, \cdots, n$, construct similar inequalities and add them together, we have
$$x_{1}^{2}+\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+\cdots+\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^{2} \leq \gamma_{1} x_{1}^{2}+\gamma_{2} x_{2}^{2}+\cdots+\gamma_{n} x_{n}^{2}$$

where each coefficient $\gamma_{k}$ is determined by
$$\gamma_{k}=\frac{a_{1}+a_{2}+\cdots+a_{k}}{k^{2} a_{k}}+\frac{a_{1}+a_{2}+\cdots+a_{k+1}}{(k+1)^{2} a_{k}}+\cdots+\frac{a_{1}+a_{2}+\cdots+a_{n}}{n^{2} a_{k}}$$

If the sequence $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ satisfies $\gamma_{k} \leq 4$ for $k=1,2, \cdots, n$, then the proof is complete. We choose $a_{k}=\sqrt{k}-\sqrt{k-1}$, then $a_{1}+a_{2}+\cdots+a_{k}=\sqrt{k}$. In this case,
$$\gamma_{k}=\frac{1}{a_{k}}\left(\frac{1}{k^{3 / 2}}+\frac{1}{(k+1)^{3 / 2}}+\cdots+\frac{1}{n^{3 / 2}}\right)$$

Notice that $\sqrt{\left(k-\frac{1}{2}\right)\left(k+\frac{1}{2}\right)}\left(\sqrt{k-\frac{1}{2}}+\sqrt{k+\frac{1}{2}}\right) \leq 2 k^{3 / 2}$, so
$$\frac{1}{k^{3 / 2}} \leq \frac{\sqrt{k+\frac{1}{2}}-\sqrt{k-\frac{1}{2}}}{\sqrt{\left(k+\frac{1}{2}\right)\left(k-\frac{1}{2}\right)}}=\frac{1}{\sqrt{k-\frac{1}{2}}}-\frac{1}{\sqrt{k+\frac{1}{2}}}$$

We get
$$\begin{array}{l} \gamma_{k}=\frac{1}{a_{k}}\left(\sum_{j=k}^{n} \frac{1}{j^{3 / 2}}\right) \leq \frac{1}{a_{k}}\left(\sum_{j=k}^{n} \frac{1}{\sqrt{j-\frac{1}{2}}}-\sum_{j=k}^{n} \frac{1}{\sqrt{j+\frac{1}{2}}}\right) \leq \frac{2}{a_{k} \sqrt{k-\frac{1}{2}}} \\ =\frac{2(\sqrt{k}+\sqrt{k-1})}{\sqrt{k-\frac{1}{2}}}<4 \end{array}$$

Note: The following similar result is left as an exercise.
$\star$ Let $x_{1}, x_{2}, \cdots, x_{n}$ be positive real numbers, prove:
$$x_{1}^{3}+\left(\frac{x_{1}+x_{2}}{2}\right)^{3}+\cdots+\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^{3} \leq \frac{27}{8}\left(x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}\right)$$