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Zeros of difference polynomials of meromorphic functions. (English) Zbl 1201.30036
Recently, studies have been made to find difference counterparts to results dealing with the value distribution of differential polynomials. As an example, a classical result due to {\it W. K. Hayman} [Ann. Math. (2) 70, 9--42 (1959; Zbl 0088.28505)], states that whenever $f$ is a transcendental meromorphic function and $a\neq 0, b$ are finite complex constants, then $f^{n}+af'-b$ has infinitely many zeros, provided $n\geq 5$, while if $f$ is entire, then $n\geq 3$ suffices. In the entire case of finite order, a difference counterpart to the Hayman result has been offered recently by {\it K. Liu} and the reviewer [Bull. Aust. Math. Soc. 81, No. 3, 353--360 (2010; Zbl 1201.30035)]. In the present paper, the author gives a difference counterpart in the meromorphic case of finite order as follows: Suppose $f$ is a transcendental meromorphic function of finite order, not of period c, and $a$ is a non-zero complex constant. Then the difference polynomial $f(z)^{n}+a(f(z+c)-f(z))-s(z)$, where $s$ is a small function with respect to $f$, has infinitely many zeros in the complex plane, provided $n\geq 8$. In two special cases, when either $f$ has a few poles only, or $f$ is the reciprocal of an entire function, the bound for $n$ will be improved. Under special assumptions, a similar result will be proved for differential polynomials of a quite general type. This paper also includes a number of concrete examples to demonstrate the importance of the assumptions of the theorems. The proofs in the paper apply standard Nevanlinna theory, partially in its difference variant established during the last decade.

30D35Distribution of values (one complex variable); Nevanlinna theory
39B32Functional equations for complex functions
Full Text: DOI
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