Uniqueness of a meromorphic function and its derivative. (English) Zbl 1056.30030

The authors consider sharing value problems of meromorphic functions and their derivatives. Let \(f(z)\) be a meromorphic function. For a positive integer \(k\) and \(a\in \mathbb{C} \cup\{\infty\}\), \(N(r,a;f\mid\geq k)\) denotes the counting function of \(a\)-points of \(f(z)\) whose multiplicities are not less than \(k\). They define \(N_p(r,a;,f) = \overline N(r,a;f) +\sum^p_{k=2}\overline N (r,a;f)\geq k)\). For two meromorphic functions \(f(z)\) and \(g(z)\), they denote by \(E_k(a;f)\) the set of all \(a\)-points whereby an \(a\)-point of multiplicity \(m\) is counted \(m\) times if \(m\leq k\) and \(k + 1\) times if \(m > k\), where \(a\in \mathbb{C}\cup \{\infty\}\) and \(k\) is a nonnegative integer or infinity. If \(E_k(a;f) = E_k(a;g)\), they say that \(f(z)\) and \(g(z)\) share the value \(a\) with weight \(k\). The authors used the symbol \((a,k)\) to express the situation of value sharing, namely, he wrote \(f(z)\) and \(g(z)\) share \((a,k)\) meaning that \(f(z)\) and \(g(z)\) share the value \(a\) with weight \(k\). It is remarked that \(f(z)\) and \(g(z)\) share value \(a\) IM (ignoring multiplicity) or CM (counting multiplicity) if and only if \(f(z)\) and \(g(z)\) share \((a,0)\) or \((a,\infty)\), respectively. Further, they define \[ \delta_p(a,f)=1-\limsup_{r\to\infty}\;\frac{N_p(r,a;f)}{T(r,f)}\,, \] where \(p\) is a positive integer.
The authors prove three theorems. I state the main theorem here. Let \(f(z)\) be a nonconstant meromorphic function and \(k\) be a positive integer. Let \(a(z)\) \((\not\equiv 0\), \(\infty\)) be a meromorphic function such that \(T(r,a) = S(r,f)\). If (i) \(a(z)\) has no zero (pole) which is also a zero (pole) of \(f(z)\) or \(f^{(k)}(z)\) with the same multiplicity, (ii) \(f(z)-a(z)\) and \(f^{(k)}(z)-a(z)\) share \((0,2)\), (iii) \(2\delta_{2+k}(0,f)+(4+k)\Theta(\infty,f) > 5+k\), then \(f(z)\) coincides with \(f^{(k)}(z)\). This result is an answer to the question posed by K.-W. Yu [JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 21, 7 p. (2003; Zbl 1021.30030)].


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


Zbl 1021.30030
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