Say we have a complex function $f(z)$ which can be represented by a power series as $f(z)=\sum^{\infty}_{n=0}a_n(z-z_0)^n.$ for $|z-z_0|<R$
Then what does it mean to say that $M(r)=sup\{f(z):|z-z_0|=r\}$ where $0<r<R$.
I understand what a supremum means in terms of a set ( i.e. the smallest upper bound of a set)
But I'm confused as to how exactly it's applicable to power series.
Is it something like $a_0$ is the first element of the set, $a_1(z-z_0)$ is the second entry in the set etc. ?
Or is it that $a_o$is the first element of the set and $a_0+a_1(z-z_0)$ is the second entry of the set ?
In fact, less is required. Being $M$ s.t. $|f(z)|\le r$ for $|z - a| = r$ is enough. Using the ML inequality: $$|a_n| = \left|\frac1{2\pi i}\int_{C(z_0,r)}\frac{f(z)}{(z - z_0)^{n+1}}dz\right|\le\frac1{2 \pi}\frac{M}{r^{n+1}} 2\pi r=\cdots$$