A Uniqueness Theorem for Meromorphic Functions Concerning Total Derivatives in Several Complex Variables
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A Uniqueness Theorem for Meromorphic Functions Concerning Total Derivatives in Several Complex Variables Ling Xu1,2 · Tingbin Cao1 Received: 6 February 2018 / Revised: 24 September 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Motivated by many differences for the total derivative between entire functions and meromorphic functions, we mainly investigate the uniqueness problems for meromorphic functions in several complex variables concerning the total derivatives. Let f and g be two nonconstant meromorphic functions on Cm , k be a positive integer such that f = 0 ⇔ g = 0, D k f = ∞ ⇔ D k g = ∞, D k f = 1 ⇔ D k g = 1. We get that if k f −1 is a nonzero constant. This is an 2δ(0, f ) + (k + 4)Θ(∞, f ) > k + 5, then DD k g−1 extension of a uniqueness theorem for entire functions due to L. Jin. There are several examples to show that our result is sharp. Keywords Meromorphic functions · Uniqueness problem · Nevanlinna theory · Total derivative · Several complex variables Mathematics Subject Classification 30D35 · 32H30
Communicated by Saminathan Ponnusamy. This research was supported by the Natural Science Foundation of China (No. 11871260, No. 11461042), the Natural Science Foundation of Jiangxi Province (No. 20161BAB201007) and the outstanding young talent assistance program of Jiangxi Province (No. 20171BCB23002) in China.
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Tingbin Cao [email protected] Ling Xu [email protected]; [email protected]
1
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China
2
School of Mathematics and Computer Sciences, Jiangxi Science and Technology Normal University, Nanchang 330013, Jiangxi, China
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L. Xu, T. Cao
1 Introduction and Main Results Let f and g be two nonconstant meromorphic functions on Cm , and let a be a finite or infinite complex number in the whole complex plane P = C1 ∪{∞}. If f −a and g −a have the same zeros with the same multiplicities, we denote it by f = a ⇔ g = a, or say that a is shared C M by f and g. If ignoring the multiplicities in above, then denote it by f = a ↔ g = a, or say that a is shared I M by f and g. A meromorphic function f on Cm is called transcendental if lim sup r →∞
T f (r ) =∞ log r
where T f (r ) is the Nevanlinna characteristic function of f (see in Sect. 2). We know that a meromorphic (entire) function f is not transcendental if and only if it is a rational function (polynomial). Uniqueness problems for meromorphic functions are a historic topic in the field of complex analysis. It is well known that two nonconstant polynomials f and g over an algebraic closed field of characteristic zero are identical if they share I M two distinct finite complex values a and b. In 1926, Nevanlinna [20] proved the well-known fivevalue theorem that if two nonconstant meromorphic functions f and g on the complex plane C1 share I M five distinct values in P, then f ≡ g. Recall that in 1976, C. C. Yang proposed a uniqueness question for functions of one complex variable: If f and g are nonconstant ent
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