Let f (z) = z ^ 2 / (z +2). Find the maximum value of | f (z) | when z varies along
disk | z | ≤ 1.
disk | z | ≤ 1.
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The function f is analytic on that disc, so by the maximum modulus theorem the maximum of |f(z)| occurs at some point on the boundary of the disc, ie, at a point where |z| = 1. But if |z| = 1 we have
|f(z)| = |z^2|/|z + 2| = 1/|z + 2|
and this is maximized when the denominator |z + 2| is minimized, which (since |z + 2| = |z - (-2)| is the distance from z to -2) is minimized when the number z is as close to the complex number -2 as possible, while still lying on the unit circle |z| = 1. This clearly happens when z = -1. So the maximum value of |f(z)| on the disk occurs when z = -1, and is
|f(-1)| = |1/(-1 + 2)| = 1.
|f(z)| = |z^2|/|z + 2| = 1/|z + 2|
and this is maximized when the denominator |z + 2| is minimized, which (since |z + 2| = |z - (-2)| is the distance from z to -2) is minimized when the number z is as close to the complex number -2 as possible, while still lying on the unit circle |z| = 1. This clearly happens when z = -1. So the maximum value of |f(z)| on the disk occurs when z = -1, and is
|f(-1)| = |1/(-1 + 2)| = 1.