doubt of intersection of ROC Z Transform dear friends of StackExchange. 
I have a doubt of the intersection of two ROC. 
I have H(Z), X(Z) and and i have to determine:  

$$ \begin{align} Y(Z)= H(Z)X(Z)\end{align}$$
$$
\displaystyle
$$

$ ROC $ H(Z)  $ \cap $ $ ROC $ X(Z)
$$
\displaystyle
$$
poles of $H(Z): p1=  \left |  -\frac{1}{4}  \right | $ ; $p2= \left |  \frac{3}{2}  \right |$ 
$$
\displaystyle
$$

The system is supposed stable. Ops, sorry i forgot to write it in the initial post, under this assumption 
the corresponding $ROC$ of $H(Z)$ is : 
$$\begin{align} \quad&amp;\ \left | \frac{1}{4} \right | &lt;z&lt;  \left | \frac{3}{2} \right | \end{align}$$
$$
\displaystyle
$$
  pole of $X(Z): p3=  \left |  2 \right |  $ $$ \displaystyle
 $$
For stability condition:
 $$\begin{align} \quad&amp;\ z&lt;  \left |2 \right | \end{align}$$
The intersection area is:
$$ \displaystyle$$

$Y(Z):  ROC $ H(Z)  $ \cap $ $ ROC $ Y(Z) = $$\begin{align} \quad&amp;\ \left | \frac{1}{4} \right | &lt; \left |  z \right | &lt; \left | \frac{3}{2} \right | \end{align}$$ 

i see it drawing the circumference and tracing the circumferences related to the poles and their corresponding ROC
 .
In order to answer this question we would need to know the ROCs of $H(z)$ and $Y(z)$. The pole radii only determine the limits of the possible ROCs. E.g., for $H(z)$ there are three different possible ROCs:

$$\begin{align}1.\quad&amp;|z|&gt;\frac{3}{2}\\
2.\quad&amp;\frac32&gt;|z|&gt;\frac14\\
3.\quad&amp;|z|&lt;\frac14\end{align}\tag{1}$$

For $Y(z)$, there are two possible ROCs:

$$\begin{align}1.\quad&amp;|z|&lt;2\\
2.\quad&amp;|z|&gt;2\end{align}\tag{2}$$

Note that each of these ROCs corresponds to a different time domain sequence.

Certain combinations of the ROCs in $(1)$ and $(2)$ overlap, others don't. Below are shown the combinations that result in an overlap of ROCs, and the resulting combined ROC. The first number is the number of the ROC of $H(z)$ given in $(1)$, the second number is the ROC of $Y(z)$ given in $(2)$:

$$\begin{align}1+2:\quad&amp;|z|&gt;2\\
1+1:\quad&amp;2&gt;|z|&gt; \frac32\\
2+1:\quad&amp;\frac32&gt;|z|&gt;\frac14\\
3+1:\quad&amp;|z|&lt;\frac14\end{align}\tag{3}$$

All other combinations of ROCs result in no overlap.