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

Abstract : Let D be the open unit disk $|x| < R$ of a complete ultrametric algebraically closed field $K$. We define the growth order $ρ(f)$, the growth type $σ(f)$ and the cotype $ψ(f)$ of an analytic function in $D$ and we show that, denoting by $ q(f, r)$ the number of zeros of f in the disk $|x| ≤ r$ and putting $|f |(r) = sup |x|≤r |f (x)|$, the infimum $θ(f)$ of the s such that $ lim r→R − q(f, r)(R − r) s = 0 $ satisfies $θ(f) − 1 ≤ ρ(f) ≤ θ(f) $ and the infimum of the $s$ such that $ lim r→R − log(|f |(r))(R − r) s = 0 $ is equal to $ρ(f)$. Moreover, if $ ψ(f) < +∞$, then $\theta(f)=\rho(f)$ and $\sigma(f)=0$. In zero residue characteristic, then $\rho(f')=\rho(f), \ \sigma(f')=\sigma(f),$ $ \psi(f')=\psi(f)$. Suppose $\K$ has zero characteristic. Consider two unbounded analytic functions $f, \ g$ in $D$. If $\rho(f)\neq \rho(g)$, then $\dsp{f\over g}$ has at most two perfectly branched values and if $\rho(f)=\rho(g)$ but $\sigma(f)\neq \sigma(g)$, then $\dsp{f\over g}$ has at most three perfectly branched values; moreover, if $2\sigma(g) < \sigma(f)$, then $\dsp{f\over g}$ has at most two perfectly branched values.
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Submitted on : Monday, November 11, 2019 - 5:22:38 PM
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Kamal Boussaf, Alain Escassut. Growth of analytic functions in an ultrametric open disk and branched values. 2019. ⟨hal-02358314⟩

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