The growth of solutions of algebraic differential equations

Walter K. Hayman

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

  • Volume: 7, Issue: 2, page 67-73
  • ISSN: 1120-6330

Abstract

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Suppose that f ⁢ z is a meromorphic or entire function satisfying P ⁢ z , f , f ′ , … , f n = 0 where P is a polynomial in all its arguments. Is there a limitation on the growth of f , as measured by its characteristic T ⁢ r , f ? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.

How to cite

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Hayman, Walter K.. "The growth of solutions of algebraic differential equations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.2 (1996): 67-73. <http://eudml.org/doc/244183>.

@article{Hayman1996,
abstract = {Suppose that \( f(z) \) is a meromorphic or entire function satisfying \( P(z, f, f', \ldots , f^\{(n)\}) = 0 \) where \( P \) is a polynomial in all its arguments. Is there a limitation on the growth of \( f \), as measured by its characteristic \( T(r, f) \)? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.},
author = {Hayman, Walter K.},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Differential equations; Entire; Meromorphic; Growth; order of meromorphic function; meromorphic solution; Nevanlinna characteristic function},
language = {eng},
month = {10},
number = {2},
pages = {67-73},
publisher = {Accademia Nazionale dei Lincei},
title = {The growth of solutions of algebraic differential equations},
url = {http://eudml.org/doc/244183},
volume = {7},
year = {1996},
}

TY - JOUR
AU - Hayman, Walter K.
TI - The growth of solutions of algebraic differential equations
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1996/10//
PB - Accademia Nazionale dei Lincei
VL - 7
IS - 2
SP - 67
EP - 73
AB - Suppose that \( f(z) \) is a meromorphic or entire function satisfying \( P(z, f, f', \ldots , f^{(n)}) = 0 \) where \( P \) is a polynomial in all its arguments. Is there a limitation on the growth of \( f \), as measured by its characteristic \( T(r, f) \)? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.
LA - eng
KW - Differential equations; Entire; Meromorphic; Growth; order of meromorphic function; meromorphic solution; Nevanlinna characteristic function
UR - http://eudml.org/doc/244183
ER -

References

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  1. BANK, S. B., On determining the growth of meromorphic solutions of algebraic differential equations having arbitrary entire coefficients. Nagoya Math. J., 49, 1973, 53-65. Zbl0268.34010MR320400
  2. BANK, S. B., Some results on Analytic and Meromorphic Solutions of Algebraic Differential Equations. Advances in Mathematics, 15, 1975, 41-61. Zbl0296.34005MR379940
  3. BANK, S. B., On the existence of meromorphic solutions of differential equations having arbitrarily rapid growth. J. reine angew. Math., 288, 1976, 176-182. Zbl0337.34007MR430371
  4. BASU, N. - BOSE, S. - VIJAYARAGHAVAN, T., A simple example for a Theorem of Vijayaraghavan. J. London Math. Soc., 12, 1937, 250-252. MR95760JFM63.0419.02
  5. BOREL, E., Mémoire sur les séries divergentes. Ann. Sci. École Norm. Sup., 16, 1899, 9-136. JFM30.0230.03
  6. GOL'DBERG, A. A., On single valued solutions of first order differential equations. Ukrain Mat. Zh., 8, 1956, 254-261 (Russian). Zbl0072.09202MR85396
  7. HAYMAN, W. K., The local growth of power series: A survey of the Wiman-Valiron method. Canad. Math. Bull., 17 (3), 1974, 317-358. Zbl0314.30021MR385095
  8. LAINE, I., Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin-New York1993. Zbl0784.30002MR1207139DOI10.1515/9783110863147
  9. STEINMETZ, N., Über das Anwachsen der Lösungen homogener algebraischer Differentialgleichungen zweiter Ordnung. Manuscripta Math., 32, 1980, 303-308. Zbl0444.34035MR595424DOI10.1007/BF01299607
  10. STEINMETZ, N., Über die eindeutigen Lösungen einer homogenen algebraischer Differentialgleichung zweiter Ordnung. Ann. Acad. Sci. Fenn., AI7, 1982, 177-188. Zbl0565.34005MR686638
  11. VALIRON, G., Fonctions Analytiques. Presses Universitaires de France, Paris1954. Zbl0055.06702MR61658
  12. VIJAYARAGHAVAN, T., Sur la croissance des fonctions définies par les équations différentielles. C. R. Acad. Sci., Paris, 194, 1932, 827-829. Zbl0004.00803
  13. WITTICH, H., Neuere Untersuchungen über eindeutige analytische Funktionen. Springer, Berlin-Göttingen-Heidelberg1955. Zbl0159.10103MR77620

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