The supremum of a sequence of functions

Question

Let $a \in \mathbb{R}_+$ and $n$ is a fixed positive integer

Let $f,g: [0,a] \to [0 , 2\pi]$ be two functions

I would like to know if is it true that $$ \sup_{n \in \mathbb{N}} \sup_{r \in [0,a]} \left| \dfrac {1} {r \cdot n^{r \cdot i \cdot f(r)}} - \dfrac {1} {r \cdot n^{r \cdot i \cdot g(r)}} \right|^2 <\infty $$ Thanks for any suggestion.

Answer 1

This is not true.

Hint:

Take $f=0,g=\pi$. Define $r_n = \frac1{\log n}$.

Then the value after the second $\sup$ in your expression will go to $\infty$ for the sequence $(n,r_n)\in \mathbb N\times [0,2\pi]$.