This is a section of an excellent paper by Granger and Morris (1976: http://www.jstor.org/stable/2345178) that shows how higher order ARMA models may be interpreted in terms of the sums of lower-order ARMA, AR and/or MA models. Some simple examples are shown in the test (such as that given below).
My question is this: How are the inequalities in 5.3 (iv) constructed - I cannot see the logic in how they are derived. Also, once one has these, how can they be used to find c and d?
It is the ARMA(1,1) with parameters $c$, $d$ and $\sigma_\xi^2$ that is given. The question is then what are the parameters of the AR(1)$+$white noise representation (if such a representation exist). This is found by solving (5.3) w.r.t. $a$, $\sigma_\eta^2$ and $\sigma_\varepsilon^2$ which leads to \begin{align} a&=c \\ \sigma_\varepsilon^2&=\left(1+d^2 - \frac{1+c^2}c d \right)\sigma_\xi^2 \\ \sigma_\eta^2 &= \frac d c \sigma_\xi^2 \end{align} The solution for $\sigma_\varepsilon^2$ is non-negative only if $1+d^2 \ge \frac{1+c^2}c d$. But the roots of the MA- and AR-polynomials in (5.2) needs to be distinct, $d\neq c$, so we must have $1+d^2 > \frac{1+c^2}c d$ equivalent to the first inequality $$ \frac 1 {1+c^2}>\frac{d}{c(1+d^2)} = \frac{\rho_1}c. $$ Similarly, the solution for $\sigma_\eta^2$ also needs to be non-negative implying that $$ \frac d c \ge 0 $$ or $$ \frac{\rho_1(1+d^2)}c \ge 0 $$ or $$ \frac{\rho_1}c \ge 0. $$
