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.
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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!!!
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 $\displaystyle z_0$ is the mean of the values on the circle $\displaystyle |z-z_0|=r$. If the mean is zero and the function is not identically 0 then it must take both positive and negative values.