Order, Type and Cotype of Growth for p-adic Entire Function - Escassut, Alain; Boussaf, Kamal; Boutabaa, Abdelbaki (Sarajevo Journal of Mathematics, 2024)
Let $\K$ be a complete ultrametric algebraically closed field and let ${\cal A}(\K)$ be the $\K$-algebra of entires functions on $\K$. For an $f\in {\cal A}(\K)$, similarly to complex analysis, one can define the order of growth as $ρ(f)=\dsp{\limsup_{r→ +∞}{\log(\log(|f|(r))\over \log r}}$. When $ρ(f) eq 0, +∞$, one can define the type of growth as $σ(f)=\dsp{\limsup_{r→ +∞}{\log(|f|(r))\over r^{ρ(f)}}}$. But here, we can also define the cotype of growth as $ψ(f)=\dsp{\limsup_{r→ +∞}{q(f,r)\over r^{ρ(f)}}}$ where $q(f,r)$ is the number of zeros of $f$ in the disk of center $0$ and radius $r$. Then we have $ρ(f)σ(f)≤ ψ(f)≤ eρ(f)σ(f)$. Moreover, if $ψ$ or $σ$ are veritable limits, then $ρ(f)σ(f)= ψ(f)$ and this relation is conjectured in the general case. Many other properties are examined concerning $ρ(f),\ σ(f),\ ψ(f)$. Particularly, we prove that if an entire function $f$ has finite order, then $\dsp{f'\over f^2}$ takes every value infinitely many times and applications are shown to branched values of meromorphic functions. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.