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 IK. 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) = 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) = 0 is equal to ρ(f). Moreover, if 0 < ρ(f) < +∞ and 0 < ψ(f) < +∞, then θ(f) = ρ(f) and σ(f) = 0. In residue characteristic zero, then ρ(f ′) = ρ(f), σ(f ′) = σ(f), ψ(f ′) = ψ(f). Suppose IK has characteristic zero. Consider two unbounded analytic functions f, g in D. If ρ(f) 6= ρ(g), then f g has at most two perfectly branched values and if ρ(f) = ρ(g) but σ(f) 6= σ(g), then f g has at most three perfectly branched values; moreover, if 2σ(g) < σ(f), then f g has at most two perfectly branched values. Subject Classification: 12J25; 30D35; 30G06