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Math Help - Maximum modulus and mean values

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    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.


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    Quote Originally Posted by dymin3 View Post
    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.
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    Thank you so much!!!
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