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In the question asked here→ Maximum Modulus Exercise

I want to know, if we just want to find maximum value of $|f(z)|$, why 'Marlu' Sir in his answer (here https://math.stackexchange.com/a/325832/168676) has done calculations to show that there are no other Maxima?

My attempt: as $f(z)=z^2-3z+2$ is analytic inside and on $|z|=1$ hence by maximum modulus theorem, maximum value of $|f(z)|$ occurs on boundary! and by traingle inequality,

$|f(z)|=|z^2-3z+2|≤|z^2|+3|z|+2$

$$≤6$$ (since on the boundary, $|z|=1$)

So from here, we know, maximum value of $|f(z)|$ cannot exceed $6$ and the fact that, at point $z=-1$ which is on boundary, $f(-1)=6$ confirms that, maximum value of $|f(z)|$ is $6$ is am I correct? Please help me...

Akash Patalwanshi
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    Some times, "find the maximum" also means to find the points where the maximum occurs. – Arthur Apr 27 '18 at 06:12
  • @Arthur sir, if we just need to find maximum value of $|f(z)|$ then is there is need of those extra calculations? Please help me.. – Akash Patalwanshi Apr 27 '18 at 06:13
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    If you only need the maximum value, and not the point(s) where it occurs, then your argument is enough: the absolute value can't be larger than $6$, and it does equal $6$ somewhere, so $6$ is the maximum. – Arthur Apr 27 '18 at 06:21
  • Thank you so much sir, for your reply, you made my day.....:-) – Akash Patalwanshi Apr 27 '18 at 06:22

1 Answers1

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No, 6 is just an upper bound but not the maximum modulus, check out the zill-complex analysis book.

Check out my annotated image if you still have issue.

Lee David Chung Lin
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KL García
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