Let $$ z(x) = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $$ where $ c > 0 $, $ x \in \mathbb{C}^{N} $, and $ P_1, P_2, P_3 \in \mathbb{C}^{M \times N} $. Is $z$ convex or not?
We can realize that the matrix $ (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3) $ is positive definite. Thus, its inverse will also be positive definite. However, when considering the whole expression $ z(x) $, I am not sure if it is convex or not.
My attempt has been to compute the second derivative of $ z $. However, the first derivative is already quite complex and I doubt this will lead to my answer.
$$ \nabla_x z(x) = x^H P^H_1 Y^{-1} P_1 - ( x^H P^H_2 Y^{-1} P_1 x x^H P^H_1 Y^{-1} P_2 + x^H P^H_3 Y^{-1} P_1 x x^H P^H_1 Y^{-1} P_3 ) $$ where $ Y = (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3) $
Any alternative way to prove convexity/non-convexity?