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 → +∞.
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Filevych, P.V. Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation. Ukrainian Mathematical Journal 53, 595–605 (2001). https://doi.org/10.1023/A:1012378721807
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DOI: https://doi.org/10.1023/A:1012378721807