$$y=\mathbb E\left[\cfrac{\mid a_1h_1w_1+\dots +a_kh_kw_k\mid ^2}{\mid a_{k+1}h_{k+1}w_{k+1}+\dots +a_nh_nw_n\mid ^2+1}\right]$$ where $h_1,\dots,h_n$ are i.i.d. zero mean circular symmetric complex Normal random variables, $a_1,\dots,a_n$ are positive valued real constants and $w_1,\dots,w_n$ are complex valued constants. The expectation is taken over all $h_1,\dots, h_n$.
I want to claim that this $y$ is maximized when we pick the numerator such that the products $a_1^2\mid w_1\mid^2,\dots,a_k^2\mid w_k\mid^2$ are all bigger than $a_{k+1}^2\mid w_{k+1}\mid^2,\dots, a_n^2\mid w_n\mid^2$. This can be easily verified for $w_i$s are real numbers since then the Numerator is the variance of the weighted sum of i.i.d. zero mean circular symmetric Normal random variables which is equal to the squared weighted sum of the variances of real and imaginary parts.
Will this hold for $w_1,\dots,w_n$ are complex valued constants?
So I proceed as follows; The numerator $$\mathbb E\mid a_1h_1w_1+\dots +a_kh_kw_k\mid ^2=\mathbb E[(a_1(h_{1R}w_{1R}-h_{1I}w_{1I})+\dots+a_k(h_{kR}w_{kR}-h_{kI}w_{kI}))^2+(a_1(h_{1R}w_{1I}-h_{1I}w_{1R})+\dots+a_k(h_{kR}w_{kI}-h_{kI}w_{kR}))^2]$$
Now I can say the Expected value of cross terms are zero so the result is $$\mathbb E\mid a_1h_1w_1+\dots +a_kh_kw_k\mid ^2=\sum_{i=1}^k a_i^2(w_{iR}^2\mathbb E(h_{iR}^2)+w_{iI}^2\mathbb E(h_{iI}^2))=\sum_{i=1}^k a_i^2(w_{iR}^2+w_{iI}^2)\sigma^2,$$ where $h_{iR}, w_{iR}$ are real parts and $h_{iI},w_{iI}$ are imaginary parts and $\sigma^2=E(h_{iI}^2)=\mathbb E(h_{iR}^2)$. Now how do I show denominator should get the smallest $a_{k+1}\mid w_{k+1}\mid^2,\dots, a_n\mid w_n\mid^2$? Please verify the above and suggest a method for denominator. Thanks.