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“2CM+1IM” Theorem for Periodic Meromorphic Functions

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Abstract

Generally, the concrete relations between two nonconstant meromorphic functions that share two values CM and one value IM are hard to determine. However, for the class \({\mathcal{F}}\) of all nonconstant meromorphic functions with the same period \({c\neq0}\), we prove a result in this paper that: let \({f(z), g(z) \in \mathcal{F}}\) such that the hyper-order \({\rho_2(f) < 1}\), if \({f(z), g(z)}\) share \({0, \infty}\) CM and 1 IM, then either \({f(z)\equiv g(z)}\) or \({f(z)=e^{az+b}g(z)}\) and \({\mu(f)=\mu(g)=1}\), where \({a=\frac{2k\pi i}{c}}\) and k is some integer. As an application of this result, we obtain an uniqueness theorem for elliptic meromorphic functions. Moreover, examples are given to illustrate that all the conditions are necessary and sharp.

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Authors and Affiliations

  1. Department of Mathematics, Fujian Normal University, Fuzhou, 350007, Fujian, People’s Republic of China

    Shengjiang Chen

  2. Department of Mathematics, Ningde Normal University, Ningde, 352100, Fujian, People’s Republic of China

    Shengjiang Chen

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  1. Shengjiang Chen
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Correspondence to Shengjiang Chen.

Additional information

This work was supported by NNSFs of China (No. 11301076, No. 11371225), NSFs of Fujian Province (No. 2011J01006, No. 2014J01004) and the Scientific Research Project of Fujian Provincial Education Department (JA15562).

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Chen, S. “2CM+1IM” Theorem for Periodic Meromorphic Functions. Results Math 71, 1073–1082 (2017). https://doi.org/10.1007/s00025-016-0555-6

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  • DOI: https://doi.org/10.1007/s00025-016-0555-6

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