Uniqueness theorem for a meromorphic function and its exact difference

Shengjiang Chen; Aizhu Xu

doi:10.4134/BKMS.b190998

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Uniqueness theorem for a meromorphic function and its exact difference

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uniqueness shared values Meromorphic exact difference

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초록

Let $f$ be a nonconstant meromorphic function of hyper order strictly less than $1$, and let $c$ be a nonzero finite complex number such that $f(z+c)\not\equiv f(z)$. We prove that if $\Delta_{c} f=f(z+c)-f(z)$ and $f$ share $0,\infty$ CM and $1$ IM, then $\Delta_{c} f= f$. Our result generalizes and greatly improves the related results.

Abstract

Let $f$ be a nonconstant meromorphic function of hyper order strictly less than $1$, and let $c$ be a nonzero finite complex number such that $f(z+c)\not\equiv f(z)$. We prove that if $\Delta_{c} f=f(z+c)-f(z)$ and $f$ share $0,\infty$ CM and $1$ IM, then $\Delta_{c} f= f$. Our result generalizes and greatly improves the related results.

참고문헌 (13)

Shengjiang Chen, "On Uniqueness of Meromorphic Functions and Their Difference Operators with Partially Shared Values", Computational Methods and Function Theory, 18(3), pp.529~536 (2018), Springer Science and Business Media LLC[DOI]
Zong-Xuan Chen, "On Sharing Values of Meromorphic Functions and Their Differences", Results in Mathematics, 63(1-2), pp.557~565 (2013), Springer Science and Business Media LLC[DOI]
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F. Gross, "Complex analysis", pp.51~67 (1976), Springer
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R. G. Halburd, "Nevanlinna theory for the difference operator", Ann. Acad. Sci. Fenn. Math, 31(2), pp.463~478 (2006)
Rodney Halburd, "Holomorphic curves with shift-invariant hyperplane preimages", Transactions of the American Mathematical Society, 366(8), pp.4267~4298 (2014), American Mathematical Society (AMS)[DOI]
W. K. Hayman, "Meromorphic Functions" (1964), Oxford Mathematical Monographs
I. Laine, "Nevanlinna theory and complex differential equations" (1993), Walter de Gruyter & Co
Ping Li, "Unicity of meromorphic functions and their derivatives", Journal of Mathematical Analysis and Applications, 285(2), pp.651~665 (2003), Elsevier BV[DOI]
Feng Lü, "Meromorphic Functions Sharing Three Values with their Difference Operators", Computational Methods and Function Theory, 17(3), pp.395~403 (2017), Springer Science and Business Media LLC[DOI]
C.-C. Yang, "Uniqueness theory of meromorphic functions" (2003), Kluwer Academic Publishers Group
Jie Zhang, "Entire functions sharing some values with their difference operators", Science China Mathematics, 57(10), pp.2143~2152 (2014), Springer Science and Business Media LLC[DOI]

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