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Extending a Theorem of Bergweiler and Langley Concerning Non-Vanishing Derivatives

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Abstract

We consider the differential operator Λk defined by

$$\Lambda_k(y) = \Psi_k(y) + a_{k-1}\Psi_{k-1}(y) +... + a_1\Psi_1(y) +a_0,$$

where a 0,…, a k−1 are analytic functions of restricted growth and Ψk(y) is a differential operator defined by Ψ1(y) = y and Ψk+1(y) = yΨk(y) + (Ψk(y))′ for k ∊ N. We suppose that k ≥ 3, that F is a meromorphic function on an annulus A(r 0), and that Λk(F) has all its zeros on a set E such that E has no limit point in A(r 0). We suppose also that all simple poles a of F in A(r0) E have res(F, a) ∉ {1,…,k − 1}. We then deduce that F is a function of restricted growth in the Nevanlinna sense. This extends a theorem of Bergweiler and Langley [1]. We show also that this result does not hold when a0,…, a k−1 are meromorphic functions.

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References

  1. W. Bergweiler and J. K. Langley, Non-vanishing derivatives and normal families, J. Anal. Math. 91 (2003), 353–367.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Bieberbach, Theorie der gewohnlichen Differentialgleichungen auf funktionentheore-tischer Grundlage dargestellt, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd LXVI. Springer-Verlag, Berlin, 1953.

    Google Scholar 

  3. J. B. Conway, Functions of One Complex Variable, Graduate Texts in Mathematics 11, Springer-Verlag, New York, 1973.

    Google Scholar 

  4. G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis 19 (1999) No.2, 173–194.

    MathSciNet  MATH  Google Scholar 

  5. W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

    Google Scholar 

  6. E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.

    MATH  Google Scholar 

  7. I, Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993.

  8. J. K. Langley, An application of the Tsuji characteristic, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991) No.2, 299–318.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Eleanor F. Clifford.

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Clifford, E.F. Extending a Theorem of Bergweiler and Langley Concerning Non-Vanishing Derivatives. Comput. Methods Funct. Theory 4, 327–339 (2005). https://doi.org/10.1007/BF03321073

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