this is my first post.

Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic sum $H$ of the $d$ variables w.r.t. the weights:

$H = \frac{1}{w_1x_1 + w_2x_2 + \cdots + w_dx_d}$.

Then, I am wondering if $H$ can be approximately expressed with some arithmetic formalization as follows:

$H \geq \sum_{i=1}^{\infty} (\tau_i(w_1, \ldots, w_d)\cdot \theta_i(x_1, \ldots, x_d)),$

where each $\tau_i$ (resp. $\theta_i$) is any function independent from the variables $x_1, x_2, \ldots, x_d$ (resp. the weights $w_1, w_2, \ldots, w_d$). The equality would hold when $w_k = x_k$ ($1\leq k \leq d$).

I could not handle this problem in my knowledge. I should be pleased to have any comments or suggestions (on the mathematical field related to this problem).

Note: I have modified the initial question as follows:

Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic sum $H$ of the $d$ variables w.r.t. the weights:

$H = \frac{1}{\frac{w_1}{x_1} + \frac{w_2}{x_2} + \cdots + \frac{w_d}{x_d}}$.

Then, I am wondering if $H$ can be approximately expressed with some arithmetic formalization as follows:

$H \geq \sum_{i=1}^{\infty} (\tau_i(w_1, \ldots, w_d)\cdot \theta_i(x_1, \ldots, x_d)),$

where each $\tau_i$ (resp. $\theta_i$) is any function independent from the variables $x_1, x_2, \ldots, x_d$ (resp. the weights $w_1, w_2, \ldots, w_d$). The equality would hold when $w_k = x_k$ ($1\leq k \leq d$).

I could not handle this problem in my knowledge. I should be pleased to have any comments or suggestions (on the mathematical field related to this problem).