Abstract
We consider the differential operator Λk defined by
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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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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DOI: https://doi.org/10.1007/BF03321073