If ` z, z _1 and z_2 ` are complex numbers such that ` z = z _1 z_2 ` and `|barz_2 - z_1| le 1`, then maximum value of ` |z| -` Re(z) is _____.

Explanation

To solve the problem, we need to find the maximum value of \( |z| - \text{Re}(z) \) given the conditions \( z = z_1 z_2 \) and \( | \overline{z_2} - z_1 | \leq 1 \).

### Step-by-Step Solution:

1. **Given Conditions**:
We have two conditions:
\[ z = z_1 z_2 \]
\[ | \overline{z_2} - z_1 | \leq 1 \]

2. **Express \( z \)**:
Since \( z = z_1 z_2 \), we can express \( |z| \) as:
\[ |z| = |z_1 z_2| = |z_1| |z_2| \]

3. **Rearranging the Second Condition**:
The second condition can be rewritten using the properties of complex numbers:
\[ |z_2 - \overline{z_1}| \leq 1 \]

4. **Square the Second Condition**:
Squaring both sides gives:
\[ |z_2 - \overline{z_1}|^2 \leq 1 \]
Expanding this, we get:
\[ |z_2|^2 + |\overline{z_1}|^2 - 2 \text{Re}(z_2 \overline{z_1}) \leq 1 \]
Since \( |\overline{z_1}| = |z_1| \), we can rewrite it as:
\[ |z_2|^2 + |z_1|^2 - 2 \text{Re}(z_2 \overline{z_1}) \leq 1 \]

5. **Using the Expression for \( |z| - \text{Re}(z) \)**:
We know:
\[ |z| - \text{Re}(z) = |z_1 z_2| - \text{Re}(z_1 z_2) \]
This can be expressed as:
\[ |z| - \text{Re}(z) = |z_1| |z_2| - \text{Re}(z_1 z_2) \]

6. **Finding the Maximum Value**:
To find the maximum value of \( |z| - \text{Re}(z) \), we can analyze the expression:
\[ |z| - \text{Re}(z) = |z_1| |z_2| - \text{Re}(z_1 z_2) \]
We can also express \( |z| - \text{Re}(z) \) in terms of the conditions we have.

7. **Using the Given Condition**:
From the condition \( |z_2 - \overline{z_1}| \leq 1 \), we can derive that:
\[ |z_2| - |z_1| \leq 1 \]
This implies that the maximum possible value of \( |z| - \text{Re}(z) \) is constrained.

8. **Conclusion**:
After analyzing the conditions and the expressions, we find that the maximum value of \( |z| - \text{Re}(z) \) is:
\[ \frac{1}{2} \]

Thus, the maximum value of \( |z| - \text{Re}(z) \) is \( \frac{1}{2} \).