Order, type and cotype of growth for p-adic entire functions: A survey with additional properties

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Let IK be a complete ultrametric algebraically closed field and let A(IK) be the IK-algebra of entire functions on IK. For an f ∈ A(IK), similarly to complex analysis, one can define the order of growth as $$\rho \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {\log |f|\left( r \right)} \right)}}{{\log r}}$$ . When ρ(f) ≠ 0,+∞, one can define the type of growth as $$\sigma \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {|f|\left( r \right)} \right)}}{{{r^\rho }\left( f \right)}}$$ . But here, we can also define the cotype of growth as $$\psi \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{q\left( {f,r} \right)}}{{{r^\rho }\left( f \right)}}$$ where q(f, r) is the number of zeros of f in the disk of center 0 and radius r. Many properties described here were first given in the Houston Journal, but new inequalities linking the order, type and cotype are given in this paper: we show that 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. Several other properties are examined concerning ρ, σ, ψ for f and f’. Particularly,we show that if an entire function f has finite order, then $$\frac{{f'}}{{{f^2}}}$$ takes every value infinitely many times.