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Derivatives of meromorphic functions and sine function. (English) Zbl 1337.30045

In his paper [Ann. Math. (2) 70, 9–42 (1959; Zbl 0088.28505)] W. K. Hayman proved that a transcendental meromorphic function \(f\) with finitely many zeros in \(\mathbb{C}\) assumes every finite non-zero value infinitely often. In this direction, the authors prove some results with new ideas. In particular, if \(f\) is a meromorphic function in the complex plane such that \(T(r;\sin z)=o(T(r; f(z))\) as \(n\to\infty\), all of whose zeros have multiplicity at least \(k+1\) (\(k\geq2\)) and all of whose poles are multiple, then \(f^{(k)}(z)-\sin z\) has infinitely many zeros.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)

Citations:

Zbl 0088.28505
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Full Text: DOI Euclid

References:

[1] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), no. 1, 9-42. · Zbl 0088.28505 · doi:10.2307/1969890
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