To prove that if both $f(z)$ and $\overline{f(z)}$ are analytic, then $f(z)$ must be constant, we can use the fact that an analytic function satisfies the Cauchy-Riemann equations and apply some properties related to complex differentiability.

Let $f(z) = u(x, y) + iv(x, y)$, where $u$ and $v$ are the real and imaginary parts of $f(z)$, respectively. Since $f(z)$ is analytic, $u$ and $v$ satisfy the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$

Since $\overline{f(z)}$ is also analytic, we can express $\overline{f(z)}$ as $\overline{f(z)} = u(x, y) - iv(x, y)$, and for this to be analytic, $u$ and $-v$ must also satisfy the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial (-v)}{\partial y} = -\frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial (-v)}{\partial x} = \frac{\partial v}{\partial x}.$

Now we have two sets of conditions:

1. From the analyticity of $f(z)$: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$
2. From the analyticity of $\overline{f(z)}$: $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}.$

Comparing these two sets of equations, we observe that they imply: $\frac{\partial v}{\partial y} = -\frac{\partial v}{\partial y} \quad \Rightarrow \quad \frac{\partial v}{\partial y} = 0$ and $\frac{\partial v}{\partial x} = -\frac{\partial v}{\partial x} \quad \Rightarrow \quad \frac{\partial v}{\partial x} = 0.$

Thus, $v$ is constant. With $v$ being constant, the Cauchy-Riemann equations imply that $u$ must also be constant, because the partial derivatives of $u$ would then be zero as well.

Therefore, $f(z) = u + iv$ is constant.

This completes the proof that if both $f(z)$ and $\overline{f(z)}$ are analytic, then $f(z)$ must be constant.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do the Cauchy-Riemann equations characterize the analyticity of a function?
2. What is the significance of a function being constant in complex analysis?
3. Can a function that is real-valued and analytic be non-constant?
4. How does the concept of harmonic functions relate to analytic functions?
5. What is the maximum modulus principle in complex analysis?

Tip: In complex analysis, analyticity is a powerful property, and the Cauchy-Riemann equations serve as a fundamental tool to determine if a function is analytic.