On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence - Filevych, P. V. (Ukrains’kyi Matematychnyi Zhurnal, 2002)
Let M f(r) and μf(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 \(_r → + ∞ M_f (r)Φ (r) = _r → + ∞ μ_f (r)Φ (r)\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞.