*I realize this matches closely with a form of the maximum modulus theorem, but my professor wants us to prove it in a non-standard way. With that said, I do wish to figure this out on my own and therefore only ask for 1 or 2 strong hints. I've been stuck on this for a long time with no clue how to start, but it's a rather important assignment and I'd feel much better if I could do this (at least mostly) on my own. So, please either give me hints in the comments or, if you wish to respond with a full answer, indicate in your answer where the hints stop and the formal proof begins. I would really appreciate it.
The problem is exactly as the title states, however my professor wants us to do the proof using a "modified" version of Cauchy's inequality, which gives a bound on the nth derivative of a function holomorphic on a disk. He wants us to modify the inequality and its proof such that it applies to any point on the disk, and not just the center. For reference, the inequality states that
|$f^{(n)}$($z_0$)|$\leq$$\frac{n!M}{R^n}$, where M is the maximum value of |f| on the disk, R is the radius of the disk, and $z_0$ is its center.
I can see several ways by which to modify this theorem, but none that are useful (as far as I can tell). How should I do this?
Additionally, he gives us the following hint: After the rework of Cauchy's inequality, consider
$g(z)$=$(f(z))^n$ for n$\in$N.
I have no idea how to use this hint or why it would even be useful.
In terms of my own attempts, the closest I've gotten was modifying Cauchy's inequality in terms of neighborhoods of points contained in the disk which are written in reference to the circle's center, and then using Taylor's theorem to try and get the modulus of the series to converge to M as I let z go to the boundary, but it didn't work out (at least with how I did it).
Please let me know what I can do! Like I said, I would greatly appreciate if hints could be kept separate from full answers. Thank you!