Can anyone help me to bound the following equation?
Here, we have $x_1+x_2+\cdots+x_k = n$
$w = \sum_{i=1}^{k}\left(c^{i-1}\left(x_i^2+2 x_i\sum_{j=i+1}^k x_j\right)\right)$
What I want to bound is $r = \sqrt{\frac{x_1^2}{w}}+\sqrt{\frac{cx_2^2}{w}}+\cdots+\sqrt{\frac{c^kx_k^2}{w}}$
For example, if $k = 3$ then $w = x_1^2+2x_1x_2+2x_1x_3+cx_2^2+2cx_2x_3+c^2x_3^2$.
$$r = \sqrt{\frac{x_1^2}{w}}+\sqrt{\frac{cx_2^2}{w}} + \sqrt{\frac{c^2x_3^2}{w}}$$
I have tried Cauchy-Schwarz inequality one the above and I can derive the following result \begin{align*} r^2 \leq \frac{3(x_1^2+cx_2^2+c^2x_3^2)}{w} \Rightarrow r\leq \sqrt{\frac{3(x_1^2+cx_2^2+c^2x_3^2)}{w}} \end{align*} However if the number of variables is $k=m$ the ratio will be approxed $\sqrt{m}$. Is it possible to give a constant upper bound for $r$ in the general case?