zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 W. K. Hayman [Ann. Math. (2) 70, 9–42 (1959; Zbl 0088.28505)], states that whenever $f$ is a transcendental meromorphic function and $a\ne 0,b$ are finite complex constants, then ${f}^{n}+a{f}^{\text{'}}-b$ has infinitely many zeros, provided $n\ge 5$, while if $f$ is entire, then $n\ge 3$ suffices. In the entire case of finite order, a difference counterpart to the Hayman result has been offered recently by 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 $f{\left(z\right)}^{n}+a\left(f\left(z+c\right)-f\left(z\right)\right)-s\left(z\right)$, where $s$ is a small function with respect to $f$, has infinitely many zeros in the complex plane, provided $n\ge 8$. 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.
MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 39B32 Functional equations for complex functions
References:
 [1] Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z + $\eta$) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008) · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1 [2] Clunie J.: On integral and meromorphic functions. J. Lond. Math. Soc. 37, 17–27 (1962) · Zbl 0104.29504 · doi:10.1112/jlms/s1-37.1.17 [3] Halburd R.G., Korhonen R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006) · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010 [4] Halburd R.G., Korhonen R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006) [5] Hayman W.K.: Picard values of meromorphic functions and their derivatives. Ann. Math. 70, 9–42 (1959) · Zbl 0088.28505 · doi:10.2307/1969890 [6] Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964) [7] Liu, K., Laine, I.: A note on value distribution of difference polynomials. Bull. Aust. Math. Soc. (to appear) [8] Laine I., Yang C.C.: Value distribution of difference polynomials. Proc. Jpn. Acad. Ser. A 83, 148–151 (2007) · Zbl 1153.30030 · doi:10.3792/pjaa.83.148 [9] Laine I., Yang C.C.: Clunie theorems for difference and q-difference polynomials. J. Lond. Math. Soc. (2) 76, 556–566 (2007) · Zbl 1132.30013 · doi:10.1112/jlms/jdm073 [10] Laine I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993) [11] Mohon’ho, A.Z.: The Nevanlinna characteristics of certain meromorphic functions, Teor. Funktsii Funktsional. Anal. i Prilozhen. 14, 83–87 (1971) (Russian) [12] Shimomura S.: Entire solutions of a polynomial difference equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, 253–266 (1981) [13] Yanagihara N.: Meromorphic solutions of some difference equations. Funkcial. Ekvac. 23, 309–326 (1980) [14] Yang C.C., Yi H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer, Dordrecht (2003)