A characterization of injective holomorphic functions in ultrametric analysis (Q1854674)

A criterion is given for the injectivity of holomorphic functions in \(p\)-adic analysis which has been conjectured by A. Escassut and M. C. Sarmant. A counter-example shows that the conjecture is incorrect for bi-analytic elements.The criterion reads as follows: A non constant holomorphic function \(f\) over an analytic subspace over \(\mathbb{C}_p\) is injective if and only if for all \(x \not = y\) \[ \left|\frac{f(x)-f(y)}{x-y}\right|^2= |f'(x) f'(y)|. \]