Order, Type and Cotype of Growth for p-adic Entire Function
Let be a complete ultrametric algebraically closed field and let A() be the -algebra of entires functions on . For an fA(), similarly to complex analysis, one can define the order of growth as ρ(f)=_r→ +∞((|f|(r))r . When ρ(f) eq 0, +∞, one can define the type of growth as σ(f)=_r→ +∞(|f|(r))r^ρ(f) . But here, we can also define the cotype of growth as ψ(f)=_r→ +∞q(f,r)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 f'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. , number=2, journal=Sarajevo Journal of Mathematics, author=Escassut, Alain and Boussaf, Kamal and Boutabaa, Abdelbaki, year=2024, month=May, pages=429–446