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Estimates on Partial Derivatives and Logarithmic Partial Derivatives of Holomorphic Functions on Polydiscs and Beyond

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Abstract

In this paper, we give estimates on partial derivatives and logarithmic partial derivatives for holomorphic functions on polydiscs. Estimates will also be utilized to characterize entire solutions of a class of partial differential equations, which gives a new form of Picard’s theorem and its higher dimensional version.

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References

  1. Ahlfors, L.V.: Conformal Invariants: Topics in Geometrical Function Theory. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  2. Biancofiore, A., Stoll, W.: Annals of Mathematical Studies. Another proof of the lemma of the logarithmic derivative in several complex variables, vol. 100, pp. 29–45. Princeton University Press, Princeton (1981)

    Google Scholar 

  3. Chiang, Y.-M., Feng, S.-J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 361, 3767–3791 (2009)

    Article  MathSciNet  Google Scholar 

  4. Davis, B.: Picard’s theorem and Brownian motion. Trans. Am. Math. Soc. 23, 353–362 (1975)

    MathSciNet  MATH  Google Scholar 

  5. Fuchs, W.H.J.: Topics in the Theory of Functions of One Complex Variable. Van Nostrand, Princeton (1967)

    MATH  Google Scholar 

  6. Goldberg, A.A., Ostrovskii, I.V.: Value Distribution of Meromorphic Functions. American Mathematical Society, Provindence (2008)

    Book  Google Scholar 

  7. Hayman, W.K.: Picard values of meromorphic functions and their derivatives. Ann. Math. 70, 9–42 (1959)

    Article  MathSciNet  Google Scholar 

  8. Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  9. Hille, E.: Ordinary Differential Equations in the Complex Domain. Dover, New York (1997)

    MATH  Google Scholar 

  10. Hiong, K.: Sur les fonctions holomorphes dont les dérivees admettent une valeur exceptionnelle. Ann Ecole Norm. Sup. 72, 165–197 (1955)

    Article  MathSciNet  Google Scholar 

  11. Krantz, S.: Function Theory of Several Complex Variables, 2nd edn. AMS Chelsea Publishing, Providence (1992)

    MATH  Google Scholar 

  12. Laine, I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyer, New York (1993)

    Book  Google Scholar 

  13. Lewis, J.L.: Picard’s theorem and Richman’s theorem by way of Harnack’s inequality. Proc. Am. Math. Soc. 122, 199–206 (1994)

    MATH  Google Scholar 

  14. Li, B.Q.: A logarithmic derivative lemma in several complex variables and its applications. Trans. Am. Math. Soc. 363, 6257–6267 (2011)

    Article  MathSciNet  Google Scholar 

  15. Li, B.Q.: Estimates on derivatives and logarithmic derivatives of holomorphic functions and Picard’s theorem. J. Math. Anal. Appl. 442, 446–450 (2016)

    Article  MathSciNet  Google Scholar 

  16. Li, B.Q.: On Picard’s theorem. J. Math. Anal. Appl. 460, 561–564 (2018)

    Article  MathSciNet  Google Scholar 

  17. Lund, M.E., Ye, Z.: Logarithmic derivatives in annuli. J. Math. Anal. Appl. 356, 441–452 (2009)

    Article  MathSciNet  Google Scholar 

  18. Nevanlinna, R.: Analytic functions, Die Grundlehren der mathematischen Wissenschaften 162. Springer, Berlin (1970)

    Google Scholar 

  19. Picard, E.: Sur une propriet è des fonctions entières. C. R. Acad. Sci. Paris 88, 1024–1027 (1879)

    MATH  Google Scholar 

  20. Picard, E.: Sur les fonctions entières. C. R. Acad. Sci. Paris 89, 662–665 (1879)

    MATH  Google Scholar 

  21. Rhoads, G., Weitsman, A.: Logarithmic derivative for minimal surfaces in $\mathbb{R}^{3}$. Comput. Methods Funct. Theory 4, 59–75 (2004)

    Article  MathSciNet  Google Scholar 

  22. Rudin, W.: Function Theory in Polydiscs. W. A. Benjamin Inc., New York (1969)

    MATH  Google Scholar 

  23. Rudin, W.: Function Theory in the Unit Ball of ${{\mathbb{C}}}^n$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241. Springer, New York (1980)

    Google Scholar 

  24. Rudin, W.: The lemma of the logarithmic derivative for subharmonic functions. Proc. Camb. Phil. Soc. 120, 347–354 (1996)

    Article  MathSciNet  Google Scholar 

  25. Segal, S.L.: Nine Introductions in Complex Analysis, Revised edn. North Holland, Elsevier (2008)

    MATH  Google Scholar 

  26. Stoll, W.: Value distribution and the Lemma of the logarithmic derivative on polydiscs. Intl. J. Math. Math. Sci. 6, 617–669 (1983)

    Article  MathSciNet  Google Scholar 

  27. Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, Oxford (1968). reprinted

    Google Scholar 

  28. Vitter, A.: The lemma of the logarithmic derivative in several complex variables. Duke Math. J. 44, 89–104 (1977)

    Article  MathSciNet  Google Scholar 

  29. Yang, L.: Value Distribution Theory. Springer, Berlin (1993)

    MATH  Google Scholar 

  30. Ye, Z.: On Nevanlinna’s second main theorem in projective space. Invent. Math. 122, 475–507 (1995)

    Article  MathSciNet  Google Scholar 

  31. Zhang, G.Y.: Curves, domains and Picard’s theorem. Bull. Lond. Math. Soc. 34, 205–211 (2002)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

    Bao Qin Li

  2. School of Mathematics, Physics and Engineering, Anhui University of Technology, Ma’anshan, 243032, People’s Republic of China

    Liu Yang

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  1. Bao Qin Li
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  2. Liu Yang
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Correspondence to Bao Qin Li.

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Li, B.Q., Yang, L. Estimates on Partial Derivatives and Logarithmic Partial Derivatives of Holomorphic Functions on Polydiscs and Beyond. J Geom Anal 31, 7333–7351 (2021). https://doi.org/10.1007/s12220-020-00490-3

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  • DOI: https://doi.org/10.1007/s12220-020-00490-3

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