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An application of hypergeometric functions to a construction in several complex variables

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Abstract

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M kG , k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for M kG were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from M kG . We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Połubiński.

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References

  1. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, MacMillan, NY, 1985.

    Google Scholar 

  2. I. I. Bavrin, A Class of Regular Bounded Functions in the Case of Several Complex Variables and Extreme Problems in that Class, Moskov Oblast. Ped. Inst., Moscow, 1976.

    Google Scholar 

  3. R. Chand and P. Singh, On certain schlicht mappings, Indian J. Pure Appl. Math. 11 (1979), 1167–1174.

    MathSciNet  MATH  Google Scholar 

  4. R. Długosz, Embedding theorems for holomorphic functions of several complex variables, J. Appl. Anal. 19 (2013), 153–165.

    MathSciNet  MATH  Google Scholar 

  5. R. Długosz and E. Lés, Embedding theorems and extreme problems for holomorphic functions on circular domains of ℂn, Complex Var. Elliptic Equ. 59 (2014), 883–899.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Dobrowolska and P. Liczberski, On some differential inequalities for holomorphic functions of many variables, Demonstratio Math. 14 (1981), 383–398.

    MathSciNet  MATH  Google Scholar 

  7. S. Fukui, On the estimates of coefficients of analytic functions, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A, 10 (1969), 216–218.

    MathSciNet  MATH  Google Scholar 

  8. I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, Inc., New York–Basel, 2003.

    MATH  Google Scholar 

  9. H. Hamada, T. Honda and G. Kohr, Parabolic starlike mappings in several complex variables, Manuscripta Math. 123 (2007), 301–324.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Higuchi, On coefficients of holomorphic functions of several complex variables, Sci. Rep. Tokyo Kyoiku Daigaku 8 (1965), 251–258.

    MathSciNet  MATH  Google Scholar 

  11. P. Liczberski, On the subordination of holomorphic mappings in ℂn, Demonstratio Math. 2 (1986), 293–301.

    MathSciNet  MATH  Google Scholar 

  12. P. Liczberski and J. Połubiński, On (j, k)-symmetrical functions, Math. Bohemica 120 (1995), 13–28.

    MathSciNet  MATH  Google Scholar 

  13. P. Liczberski and J. Połubiński, Symmetrical series expansion of complex valued functions, New Zealand J. Math. 27 (1998), 245–253.

    MathSciNet  MATH  Google Scholar 

  14. P. Liczberski and J. Połubiński, Functions (j, k)-symmetrical and functional equations with iterates of the unknown function, Publ. Math. Debrecen 60 (2002), 291–305.

    MathSciNet  MATH  Google Scholar 

  15. P. Liczberski and J. Połubiński, A uniqueness theorem of Cartan–Gutzmer type for holomorphic mappings in Cn, Ann. Polon. Math. 79 (2002), 121–127.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Michiwaki, Note on some coefficients in a starlike functions of two complex variables, Res. Rep. Nagaoka Tech. College 1 (1963), 151–153.

    MathSciNet  Google Scholar 

  17. J. A. Pfaltzgraff and T. J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Sklodowska, Sect. Math. 53 (1999), 193–207.

    MathSciNet  MATH  Google Scholar 

  18. W. Rudin, Functional Analysis, McGraw-Hill Inc., New York, 1991.

    MATH  Google Scholar 

  19. K. Sakaguchi, On certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72–75.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Stankiewicz, Some remarks on functions starlike with respect to symmetric points, Ann. Univ. Mariae Curie-Sklodowska, Sect. Math. 19 (1965), 53–59.

    MathSciNet  MATH  Google Scholar 

  21. A. Temljakov, Integral representation of functions of two complex variables, Izv. Acad. Sci. SSSR, Ser. Math. 21 (1957), 89–92.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Łódź University of Technology, Zwirki 36, 90-924, Łódź, Poland

    Renata Długosz & Piotr Liczberski

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  1. Renata Długosz
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  2. Piotr Liczberski
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Correspondence to Piotr Liczberski.

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Długosz, R., Liczberski, P. An application of hypergeometric functions to a construction in several complex variables. JAMA 137, 707–721 (2019). https://doi.org/10.1007/s11854-019-0012-z

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  • DOI: https://doi.org/10.1007/s11854-019-0012-z

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