Abstract
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 \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function f, it is necessary and sufficient that ln Φ′(x) = o(Φ(x)) as x → +∞.
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REFERENCES
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Filevych, P.V. On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence. Ukrainian Mathematical Journal 54, 1386–1392 (2002). https://doi.org/10.1023/A:1023443926292
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DOI: https://doi.org/10.1023/A:1023443926292