Let $N$, $D$, $P\in\mathbb{R}^{n}$ be arbitrary vectors, for which all elements of $N$ and $D$ are $\geq0$ and all elements of $P$ are $>0$. Specifically $P = [p^{(n-1)}\quad p^{(n-2)}\quad \cdots\quad p\quad 1]$, with $p>0$ considered to be known. Also, $||N||>0$ and $||D||>0$, with the elements of $N$ and $D$ being unknown. Additionally, $A= \left[ {\begin{array}{cccc} (aj)^{(n-1)} & 0 & \dots & 0 \\ 0 & (aj)^{(n-2)} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \\ \end{array} } \right] $ is an $n\times n$ diagonal matrix, with $a$ being a positive scalar $a>0$ which we can manipulate as we wish (it is not unknown, instead it is a design parameter), and $j$ is the imaginary element for which $j^{2}=-1$. Now consider the two norms $\left| \frac{N\cdot (PA)}{D\cdot (PA)}\right|$ (or equally $\left| \frac{NAP^T}{DAP^T}\right|$) and $\left| \frac{N\cdot P}{D\cdot P}\right|$. I am interested in finding a number $K$ (it can also be a function of $a$, that is $K(a)$) and maybe a value for the parameter $a$, so as it holds: \begin{equation} \left| \frac{N\cdot (PA)}{D\cdot (PA)}\right|\leq K \times \left| \frac{N\cdot P}{D\cdot P}\right| \end{equation}
It is noted that the symbol $(\cdot)$, denotes the inner product between vectors, and $\times$ is the simple multiplication. Additionally, wherever the symbol is omitted, a simple vector to matrix multiplication is implied.
This interesting problem is the key idea for constructing a complicated robust control design for a system with unstable poles, which I deal with in the context of my thesis. I can give more details if needed. Thank you all for your time.
With arbitrary vectors I assume the three vectors could be anything point in $\mathbb{R}^n$. Though, due to the problem formulation one could factor out the norm of each vector and cancel them on both the sides of the inequality. So instead each vector is assumed to be of unit length.
The worst case scenario for the inequality would be when $N \cdot P=0$. If the left hand side of your inequality then does not have to also be equal to zero then $K$ is not well defined/infinite. Due to your definition of $A$ for $n=1$ it holds that $N \cdot (P A) = N \cdot P$, similar for the dot products with $D$ and therefore in that case $K=1$. For $n>1$ in general $N \cdot (P A) \neq N \cdot P$ and thus there should exists $N,P\mathbb{R}^n$ for which $N \cdot P=0$ and $N \cdot (P A) \neq 0$ ($D$ has to just be chosen such that $D \cdot (P A)$ is bounded, but this is the case when only considering unit vectors). Therefore, when $n>1$ $K$ is not well defined/infinite.