My question concerns the following extract, taken from page 881 of this paper by Muir and Chipman. The paper presents a three-stage implicit Runge-Kutta method with parameters $c_2$ and $c_3$.
In an attempt to obtain the requirements for A-stability, highlighted in green at the bottom of the image, I set out trying to find conditions on the parameters $\alpha,\beta$ and $c_2$ for which $\vert R(z) \vert \leq 1$, where $z$ is any point on the imaginary axis. This led me to the condition $a(a+2b-6bc)\leq 0$. This is starting to look a little more like the second condition highlighted in the text, however, I am still not sure about:
why the conditions given in the text are strict inequalities,
where the condition $\beta/\alpha > 0$ comes from.
Now, of course, in the definition of A-stability, we also have to check that the stability function is analytic in $\mathbb{C}^-$, so perhaps this is where $\beta/\alpha > 0$ comes from? It isn't immediately obvious, as the cubic polynomial on the denominator is difficult to handle when identifying the poles of this function.
Any help resolving the above points is, as always, much appreciated.