Extending a Theorem of Bergweiler and Langley Concerning Nonvanishing Derivatives

We consider the differential operator Λk defined by Λk(y) = Ψk(y) + ak−1Ψk−1(y) + . . .+ a1Ψ1(y) + a0 , where a0, . . . , ak−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(r0), and that Λk(F ) has all its zeros on a set E such that E has no limit point in A(r0). 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 for a0, . . . , ak−1 meromorphic functions. 1 Mathematics Subject Classification (1991): 30D30, 30D35