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On the growth of universal functions

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Abstract

In this paper, we study the growth of universal functions (Taylor series) on the unit discD. We describe a class of growth types prohibited for such a function. By investigating the relation between growth and value distribution, we prove that every universal function assumes every complex value, with at most one exception, on sequences inD that approach ϖD rather slowly. We use this to get a large class of equations, including polynomials with Nevanlinna coefficients and equations of iterates, that no universal function satisfies. Finally, we produce a universal function whose growth is bounded by

$$\exp (\exp (\frac{M}{{1 - \left| z \right|}}\log \log \frac{4}{{1 - \left| z \right|}})),$$

, which is close to the rates of growth prohibited.

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References

  1. C. Chui and M. N. Parnes,Approximation by overconvergence of power series, J. Math. Anal. Appl.36 (1971), 693–696.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Costakis and A. Melas,On the range of universal Taylor series, submitted.

  3. W. K. Hayman,Meromorphic Functions, Oxford University Press, 1964.

  4. S. Hellerstein and L. A. Rubel,Subfields that are algebraically closed in the field of all meromorphic functions, J. Analyse Math.12 (1964), 105–111.

    MATH  MathSciNet  Google Scholar 

  5. J.-P. Kahane,Trois papiers sur les méthodes de Baire en analyse harmonique, Prép. 97-25, Univ. de Paris-Sud Math., Bât. 425, 91405 Orsay, France.

  6. E. S. Katsoprinakis and M. Papadimitrakis,Extensions of a theorem of Marcinkiewicz-Zygmund and of Rogosinski's formula and an application to universal Taylor series, Proc. Amer. Math. Soc.127 (1999), 2083–2090.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Luh,Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen88 (1970).

  8. W. Luh,Universal approximation properties of overconvergent power series on open sets Analysis6 (1986), 191–207.

    MATH  MathSciNet  Google Scholar 

  9. A. Melas, V. Nestoridis and I. Papadoperakis,Growth of coefficients of universal Taylor series and comparison of two classes of functions, J. Analyse Math.73 (1997), 187–202.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Melas and V. Nestoridis,Universality of Taylor series as a generic property of holomorphic functions, submitted.

  11. V. Nestoridis,Universal Taylor series, Ann. Inst. Fourier (Grenoble)46 (1996), 1293–1306.

    MATH  MathSciNet  Google Scholar 

  12. V. Nestoridis,An extension of the notion of universal Taylor series, Proceedings CMFT'97, Nicosia, Cyprus, Oct. 1997.

  13. R. Nevanlinna,Analytic Functions, Springer-Verlag, Berlin-Heidelberg, 1970.

    MATH  Google Scholar 

  14. I. I. Privalov,Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Math. Ann.93 (1924), 149–152.

    Google Scholar 

  15. W. Rudin,Real and Complex Analysis, McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  16. A. Zygmund,Trigonometric Series, second edition reprinted, Vol. I, II, Cambridge University Press, 1979.

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Authors and Affiliations

  1. Department of Mathematics, University of Athens, Panepistimiopolis, 157-84, Athens, Greece

    Antonios D. Melas

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  1. Antonios D. Melas
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Correspondence to Antonios D. Melas.

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The author would like to thank the referee for suggesting references [3] and [4].

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Melas, A.D. On the growth of universal functions. J. Anal. Math. 82, 1–20 (2000). https://doi.org/10.1007/BF02791219

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  • DOI: https://doi.org/10.1007/BF02791219

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