# Thread: Maximum modulus and mean values

1. ## Maximum modulus and mean values

Let F be analytic and non-constant on the disc |z - z0| < R, and suppose that Re(F(z0)). Show that on every circle |z - z0| = r, 0 < r < R, Re(F) assumes both positive and negative values.

I do not know how to solve this

Plz help me!!!

2. Originally Posted by dymin3
Let F be analytic and non-constant on the disc |z - z0| < R, and suppose that Re(F(z0)). Show that on every circle |z - z0| = r, 0 < r < R, Re(F) assumes both positive and negative values.
The real part of an analytic function is a harmonic function. The mean value theorem for harmonic functions says that the value of the function at $z_0$ is the mean of the values on the circle $|z-z_0|=r$. If the mean is zero and the function is not identically 0 then it must take both positive and negative values.

3. Thank you so much!!!