## Abstract

In this paper, we describe a family of meromorphic functions in \(\mathbf {C}\) from analyzing some properties of these *L*-functions in the extended Selberg class and show two uniqueness results of such a function, in terms of shared values with a general meromorphic function in \(\mathbf {C}\). In particular, we show the condition “\(\displaystyle {1\mathsf{CM}+3\mathsf{IM}}\) value-sharing” suffices.

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## Notes

When \(\rho (f)=0\), we do not need to consider \(\tau (f)\) in view of the transcendentality of

*f*.Recall when

*f*has finite order, then*S*(*r*,*f*) is \(\displaystyle {O\left( \log r\right) }\) as a straightforward consequence of the logarithmic derivative lemma.The same observation of \(\tau (f)\) as mentioned in Theorem 2.1 applies here when \(\rho (f)=0\).

## References

Adams, W., Straus, E.: Non-archimedian analytic functions taking the same values at the same points. Illinois J. Math.

**15**, 418–424 (1971)Garunkštis, R., Grahl, J., Steuding, J.: Uniqueness theorems for \(L\)-functions. Comment. Math. Univ. St. Pauli

**60**, 15–35 (2011)Gonek, S., Haan, J., Ki, H.: A uniqueness theorem for functions in the extended Selberg class. Math. Z.

**278**, 995–1004 (2014)Gundersen, G.G.: Meromorphic functions that share three values IM and a fourth value CM. Complex Variables Theory Appl.

**20**, 99–106 (1992)Hayman, W.K.: Meromorphic functions. Oxford University Press, Oxford (1964)

Hu, P., Li, B.Q.: A simple proof and strengthening of a uniqueness theorem for \(L\)-functions. Canad. Math. Bull.

**59**, 119–122 (2016)Li, B.Q.: A result on value distribution of \(L\)-functions. Proc. Amer. Math. Soc.

**138**, 2071–2077 (2010)Li, X., Yi, H.: Results on value distribution of \(L\)-functions. Math. Nachr.

**286**, 1326–1336 (2013)Mues, E.: Meromorphic functions sharing four values. Complex Variables Theory Appl.

**12**, 169–179 (1989)Nevanlinna, R.: Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen. Acta Math.

**48**, 367–391 (1926)Nevanlinna, R.: Analytic functions. Springer-Verlag, New York-Berlin (1970)

Ozawa, M.: Unicity theorems for entire functions. J. Analyse Math.

**30**, 411–420 (1976)Selberg, A.: Old and new conjectures and results about a class of Dirichlet series. Proceedings of the Amalfi conference on analytic number theory (Maiori, 1989), 367–385. University of Salerno, Salerno, (1992)

Steinmetz, N.: Reminiscence of an open problem: remarks on Nevanlinna’s four-value-theorem. Southeast Asian Bull. Math.

**36**, 399–417 (2012)Steuding, J.: Value-distribution of \(L\)-functions. Springer, Berlin (2007)

Ueda, H.: Some estimates for meromorphic functions sharing four values. Kodai Math. J.

**17**, 329–340 (1994)Yang, C.C., Yi, H.: Uniqueness theory of meromorphic functions. Kluwer Academic Publishers, Dordrecht (2003)

Yang, L.: Value distribution theory. Springer-Verlag, Berlin (1993)

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Dedicated to Professor Seiki Mori on the occasion of his 72nd birthday.

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Han, Q. Some uniqueness results related to *L*-functions.
*Boll Unione Mat Ital* **10**, 503–515 (2017). https://doi.org/10.1007/s40574-016-0081-1

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DOI: https://doi.org/10.1007/s40574-016-0081-1