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On the growth of entire and meromorphic functions of infinite order. (English) Zbl 0722.30016
Let $$\phi$$ (x) be a positive increasing function satisfying $$\int^{\infty}dx/\phi (x)<\infty$$. It is shown that if f is a meromorphic function, then there exists a set E of logarithmic density 1 such that $\lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))\log (\phi (\log T(r)))}=0.$ Here M(r) and T(r) denote the maximum modulus and Nevanlinna characteristic of f. If f is entire, then $\lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))}=0.$ The proof depends on the Poisson-Jensen formula, a growth lemma for real functions, and - in the case that f is meromorphic - Cartan’s lemma. Next it is shown that the condition that $$\int^{\infty}dx/\phi (x)<\infty$$ cannot be weakened. To construct examples which show this, the authors first use conformal mappings of strips to construct a function which is harmonic in the plane except on certain curves where it is subharmonic or superharmonic. Then they approximate this function by log$$| f|$$ where f is meromorphic and has zeros (respectively poles) on these curves. The estimation of the asymptotic behavior of the conformal mapping of the strip seems to require some regularity hypothesis on $$\phi$$ (x) which is not stated in the paper, but will be given in a forthcoming correction (same journal, to appear).

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D15 Special classes of entire functions of one complex variable and growth estimates 30D20 Entire functions of one complex variable, general theory 30D30 Meromorphic functions of one complex variable, general theory
##### Keywords:
infinite order; maximum modulus; Nevanlinna characteristic
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