I Demonstration of inequality between 2 variance expressions

AI Thread Summary
The discussion centers on proving the inequality between two variance expressions, specifically that σ_o,1² < σ_o,2². The first observable variance σ_D,1² is defined as a sum involving the terms (2ℓ + 1) and a constant factor, while the second observable variance σ_D,2² incorporates a similar summation but with a different formulation. The goal is to establish the inequality by demonstrating that the squared sum of C_ℓ is greater than a specific combination of terms involving X and Y, where X is an increasing function and Y is a decreasing function of ℓ. The author is seeking suggestions or assistance to complete this proof. The discussion highlights the mathematical intricacies involved in comparing these variance expressions.
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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.
 
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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
 
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