I am sorry for the inconvenience to change the initial query (whose solution must be collect with yours). I was happy to have your helpful comment on the problem which troubles me.

It should be better to modify my question as follows. Suppose that $H$ is written as $\frac{1}{\frac{w_1}{x_1} + \cdots + \frac{w_d}{x_d}}$. Then, is it possible to express $H$ in an arithmetic form like $H \geq \Sigma_i (\tau_i(w_1, \ldots, w_d)\cdot \theta_i(x_1, \ldots, x_d))$ with equality $w_k = x_k (1\leq k \leq d)$, where $\tau_i$ (resp. $\theta_i$) is any function independent from the variables $x_1, \ldots, x_d$ (resp. the weights $w_1,\ldots, w_d$)?

Thank you very much. That's very helpful for me. And now, I notice that I should have written the equality condition as $w_k = \frac{1}{x_k}$, not $w_k = x_k$. I am sorry for that. My concern is that though the Schwarz inequality holds, when we consider $w_k = \frac{1}{x_k}$ as the expected equality condition, $H$ does not reach the right term; $\frac{1}{\sqrt{\Sigma_i w_i^2}}\frac{1}{\sqrt{\Sigma_i x_i^2}}$.