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Math Help - complex: question with application of Cauchy's Inequality

  1. #1
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    complex: question with application of Cauchy's Inequality

    Please help with this problem. Thank you.

    Let f be amalytic in D = {z in C | |z| <= 10} with f(1) = 0, f"(0) = -i. Prove that there exists a z0 in D such that |f(z0)|= sqrt2009.
    Last edited by conmeo; December 1st 2009 at 11:55 PM.
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  2. #2
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    Can we not argue on continuity grounds that the maximum modulus function of a non-constant, analytic function is a continuous, monotonic, and increasing function of the outward radius? And since f(1)=0 and f''(0)=-i, f(z) is not a constant function and by Cauchy's inequality for the second derivative, |f''(\gamma)|\leq \frac{2}{R^2}M where \gamma is the contour z=10e^{it} and |f(\gamma)|\leq M. This then would imply 50\leq M. Therefore, the maximum of |f(z)| on \gamma must be larger than 50 and f(1)=0 so that the maximum modulus between r=1 and r=10 is a continuous function of r at least in the range of 1<\sqrt{2009}<50.

    Also, I thought it interesting to verify this directly with the function f(z)=1/2 i z^3-i/2 z^2 which satisfies f(1)=0 and f''(0)=-i in which case the maximum modulus as a function of r is r^2/2(r+1).
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