Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation

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 l(r) be a continuously differentiable function convex with respect to ln r. We establish that, in order that ln M _f(r) ∼ ln μ _f (r), r → +∞, for every entire function f such that μ _f (r) ∼ l(r), r → +∞, it is necessary and sufficient that ln (rl′(r)) = o(l(r)), r → +∞.