On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Abstract

Let M _f(r) and μ_f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that x = o(Φ(x)) as x → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞.