Given that $z_1, z_2, z_3$ are three unimodular complex numbers, it implies that $|z_1| = |z_2| = |z_3| = 1$. We need to determine the maximum possible value of $|z|$, where

$z = (z_1 + z_2 + z_3) \left( \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right).$

Step-by-Step Analysis:

1. Since $|z_1| = |z_2| = |z_3| = 1$, it follows that for any $z_i$, $\frac{1}{z_i} = \overline{z_i}$ (the complex conjugate of $z_i$) because:

$\frac{1}{z_i} = \frac{\overline{z_i}}{|z_i|^2} = \overline{z_i},$

where $|z_i| = 1$.

2. Therefore, we can express $z$ as:

$z = (z_1 + z_2 + z_3) \left( \overline{z_1} + \overline{z_2} + \overline{z_3} \right).$

3. Notice that $\overline{z_1} + \overline{z_2} + \overline{z_3} = \overline{z_1 + z_2 + z_3}$, so:

$z = (z_1 + z_2 + z_3) \cdot \overline{(z_1 + z_2 + z_3)}.$

4. This expression simplifies to:

$z = |z_1 + z_2 + z_3|^2.$

We need to determine the maximum value of $|z|$, which is $|z| = |z_1 + z_2 + z_3|^2$.

Maximizing $|z_1 + z_2 + z_3|$:

The maximum value of $|z_1 + z_2 + z_3|$ occurs when $z_1, z_2, z_3$ are aligned in the same direction on the unit circle (i.e., they have the same argument). In that case:

$|z_1 + z_2 + z_3| = |1 + 1 + 1| = 3.$

Thus, the maximum value of $|z|$ is:

$|z| = 3^2 = 9.$

Conclusion:

The value of $|z|$ cannot exceed 9.

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Would you like more details or have any other questions?

Related Questions:

1. What is the geometric interpretation of unimodular complex numbers?
2. How do you compute the argument of a complex number?
3. What are some properties of the modulus of a complex number?
4. Can this problem be generalized to more than three complex numbers?
5. How does the sum of complex numbers on the unit circle behave?

Tip:

When working with complex numbers on the unit circle, remember that their magnitude is always 1, which simplifies many calculations involving their sum and product.