Demonstration of inequality between 2 variance expressions

[fab13]

TL;DR

In an astrophysics context, I would like to prove than $\sigma_{o, 1}^{2}<\sigma_{o, 2}^{2}$ but I have difficulties to derive this inequality.

Just to remind, $C_\ell$ is the variance of random variables $a_{\ell m}$ following a Gaussian PDF (in spherical harmonics of Legendre) :

$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)$

1) Second observable :
$\sigma_{D, 2}^{2}=\dfrac{2 \sum_{\ell_{\min }}^{\ell_{\max }}(2 \ell+1)}{\left(f_{s k y} N_{p}^{2}\right)}$
so :
$\sigma_{o, 2}^{2}=\dfrac{\sigma_{D, 2}^{2}}{\left(\sum_{\ell_{\min }}^{\ell_{\max }}(2 \ell+1) C_{\ell}\right)^{2}}$

2) First observable :
$\sigma_{D, 1}^{2}=\sum_{\ell_{\min }}^{\ell_{\max }} \dfrac{2}{(2 \ell+1)\left(f_{s k y} N_{p}^{2}\right)}$
so :
$\sigma_{o, 1}^{2}=\dfrac{\sigma_{D, 1}^{2}}{\left(\sum_{\ell_{\min }}^{\ell_{\max }} C_{\ell}\right)^{2}}$
3) Goal :
I would like to prove than $\sigma_{o, 1}^{2}<\sigma_{o, 2}^{2}$ but I have difficulties to derive this inequality.

 

[fab13]

Things are progressing in my demonstration.

All I need to do now is to prove the following inequality, by taking $X = 2 \ell + 1$ and $Y = C_\ell$:

$\big(\sum Y \big)^{2} > \sum X^{- 1} \sum XY^{2}\quad (1)$

with $X$ and $Y$ which are functions of $\ell$ (see above) and $X$ is increasing while $Y$ is assumed to be decreasing.

The sum $\sum$ is actually done over $\sum_{\ell =\ell_{min}}^{\ell_{max}}$, it was just to make it more readable than I did not write in $(1)$.

Any suggestion, track or help is welcome.

Best regards