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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
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[1] W. Al-Katifi,On the asymptotic values and paths of certain integral and meromorphic functions, Proc. London Math. Soc. (3)16 (1966), 599–634. · Zbl 0145.30801 · doi:10.1112/plms/s3-16.1.599
[2] V. F. Azarin,The asymptotic behavior of subharmonic and entire functions, Dokl. Akad. Nauk SSSR229 (1976), 1289–1291. · Zbl 0352.31001
[3] W. Bergweiler,Maximum modulus, characteristic and area on the sphere, preprint (1990). · Zbl 0703.30025
[4] C. T. Chuang,Sur la croissance des fonctions, Kexue Tongbao26 (1981), 677–684. · Zbl 0496.30019
[5] W. H. J. Fuchs,Topics in Nevanlinna theory, Proc. of the NRL Conference on Classical Function Theory, 1970. · Zbl 0294.30021
[6] N. V. Govorov,On Paley’s problem, Funkts. Anal. Prilozh.3 (1969), 35–40.
[7] W. K. Hayman,Meromorphic Functions, Oxford, 1964.
[8] W. K. Hayman,Subharmonic Functions, Vol. 2, Academic Press, London, 1989. · Zbl 0699.31001
[9] T. Kövari,A gap theorem for entire functions of infinite order, Michigan Math. J.12 (1965), 133–140. · Zbl 0152.06604 · doi:10.1307/mmj/1028999302
[10] I. I. Marchenko and A. I. Shcherba,Growth of entire functions, Siberian Math. J. (Engl. Transl.)25 (1984), 598–606. · Zbl 0581.30025 · doi:10.1007/BF00968899
[11] R. Nevanlinna,Analytic Functions, Springer, New York, 1974. · Zbl 0014.16304
[12] V. P. Petrenko,The growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR33 (1969), 414–454(in Russian). · Zbl 0194.11001
[13] G. Valiron,Sur un theorème de M. Wiman, in: Opuscula Math. A. Wiman dedicata, 1930, pp. 1–12.
[14] A. Wahlund,über einen Zusammenhang Zwischen dem Maximalbetrage der ganzen Funktion und seiner unteren grenze nach dem Jensensche Theoreme, Arkiv Math.21A (1929), 1–34. · JFM 55.0775.01
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