Let $L\subseteq\mathbb{C}\_p$ be a finite extension of $\mathbb{Q}\_p$, $r$ be a positive real number, and $f$ be a series $\sum\_{n\in \mathbb{Z}} a\_nz^n$ convergent in $D= \{x\in \mathbb{C}\_p|0<v(x)\leq r \}$ where $a\_n$ are elements in $L$. Then I want to know if the following are equivalent.


(1) $f$ is a bounded function in the metric of $\mathbb{C}\_p$


(2) $f$ only has finitely many zeros in $D$


(3) the set $\{\lvert a\_n\rvert\}$ is bounded as a subset of $\mathbb{R}$(in the Euclid metric)


Symbols: $\lvert a\_n\rvert\mathrel{:=}p^{-v(a\_n)}$ and $v$ is the valuation of $\mathbb{C}\_p$ extended by the valuation on $\mathbb{Q}\_p$.


Motivations: I want to use this to prove some properties of the Robba ring over $L$, e.g., $\varepsilon^\dagger$ is a field.


Thanks!



$p$-adic series bounded if and only if it has finitely many zeros
(2) and (3) are equivalent. This is corollary 3.3 in Laurent Berger's IHP course notes [Galois representations and $(\varphi, \Gamma)$-modules](http://perso.ens-lyon.fr/laurent.berger/autrestextes/CoursIHP2010.pdf) in 2010. In the same way, we can prove (1) and (3) are equivalent (in one direction, convergence is used).


