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A note on the value distribution of \(f^1(f^{(k)})^n\). (English) Zbl 1355.30030

The value distribution of \(f^{l}(f^{(k)})^{n}\) is discussed and the following result is proved: Let \(f\) be a transcendental meromorphic function in \(\mathbb{C}\), let \(l>1\), \(n >1\), \(k >1\) be integers, and let \(a\) be a nonzero constant. Then \[ (l - 1)T(r, f) \leq N\big(r, 0; f^{l}(f^{(k)})^{n} - a\big) + S^{*}(r, f), \] where \(S^{*}(r, f) = o\{T(r, f)\}\) as \(r \to \infty\), possibly outside a set of logarithmic density zero.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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