Growth of analytic functions in an ultrametric open disk and branched values

Abstract

Let $D$ be the open disk $|x|< R$ of a complete ultrametric algebraically closed field $\mathbb K$. We define the growth order $\rho(f)$, the growth type $\sigma(f)$ the cotype $\psi(f)$ and another expression $\theta(f)$ of an analytic function in $D$ and we show relations between the number of zeros of $f$ and the ``maximum modulus'' of $f$ involving $\rho,\sigma,\psi$. Then $\rho(f)$ lies in $[\theta(f)-1, \theta(f)]$. Moreover, if $0< \rho(f)< +\infty$ and $0< \psi(f)< +\infty$, then $\theta(f)=\rho(f)$ and $\sigma(f)=0$. Suppose $\mathbb K$ has characteristic zero and consider two unbounded analytic functions $f,g$ in $D$. The number of perfectly branched values admits overbounds linked to $\rho(f), \rho(g), \sigma(f), \sigma(g)$. In residue characteristic zero, then $\rho(f')=\rho(f),\sigma(f')=\sigma(f),$ $ \psi(f')=\psi(f)$, which can be applied to Levi-Civta fields.