I would like to show that when a circular disk $|z| \leq \rho$ is translated one unit to the right, the point of maximum modulus in the resulting disk $|z+1| \leq \rho$ is $ z = 1 + \rho.$ Any hint or a proof for this?
Point of maximum modulus of a circular disk
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complex-analysis
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0One unit to the left maybe? The point of maximum modulus in the disk you gave is $z=-1-\rho$. – 2012-11-05
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0Can you elaborate on this? – 2012-11-05
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0If you move the first disk one unit to the right, the equation is $|z-1| \leq \rho$. – 2012-11-05
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0Another way to see this is to consider $|z+1|$ if $z=1+\rho$. Then $|z+1| = 2+\rho$, and clearly, if $\rho\geq 0$, we cannot have $2+\rho \leq \rho$. So, you need to change the disk or the point of maximum modulus. – 2012-11-05