Computational Methods and Function Theory

, Volume 17, Issue 1, pp 19–45 | Cite as

On the Characterisations of a New Class of Strong Uniqueness Polynomials Generating Unique Range Sets

  • Abhijit Banerjee
  • Sanjay Mallick


The purpose of the paper is to introduce a new class of strong uniqueness polynomials with the best possible answer to the question posed by the first author (Banerjee, Ann Acad Sci Fenn Math 40:465-474, 2015). We find the corresponding unique range set which improves a result due to Frank–Reinders (Complex Var Theory Appl 37(1):185–193, 1998). At the time of characterising the strong uniqueness polynomial, we also encounter the cases where the polynomial is not necessarily critically injective and find its degree and the corresponding unique range set. Furthermore, we find some connections between this new class of polynomials and most of the polynomials generating URSM introduced so far in the literature of value distribution theory. As an application of our results, we will show that the theorems improve many previous results. Lastly, we rectify the gap in the last section of An (Acta Math Vietnam 27(3), 251–256, 2002, p. 255, l. 11–14) and thus improve the result obtained by An (Acta Math Vietnam 27(3), 251–256, 2002) significantly.


Meromorphic function Uniqueness Strong uniqueness polynomial Unique range set 

Mathematics Subject Classification




This research work is supported by the Council of Scientific and Industrial Research, Extramural Research Division, CSIR Complex, Pusa, New Delhi 110012, India, under project no. 25(0229)/14/EMR-II. The authors wish to thank the referee for his/her valuable suggestions towards improvement of the paper.


  1. 1.
    An, T.T.H.: A new class of unique range sets for meromorphic functions on \(\mathbb{C}\). Acta Math. Vietnam. 27(3), 251–256 (2002)MathSciNetMATHGoogle Scholar
  2. 2.
    Banerjee, A.: Uniqueness of meromorphic functions sharing two sets with finite weight II. Tamkang J. Math. 41(4), 379–392 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Banerjee, A.: A new class of strong uniqueness polynomials satisfying Fujimoto’s condition. Ann. Acad. Sci. Fenn. Math. 40, 465–474 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Banerjee, A., Lahiri, I.: A uniqueness polynomial generating a unique range set and vise versa. Comput. Method Funct. Theory 12(2), 527–539 (2012)CrossRefMATHGoogle Scholar
  5. 5.
    Fujimoto, H.: On uniqueness of meromorphic functions sharing finite sets. Am. J. Math. 122, 1175–1203 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fujimoto, H.: On uniqueness polynomials for meromorphic functions. Nagoya Math. J. 170, 33–46 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Frank, G., Reinders, M.: A unique range set for meromorphic functions with 11 elements. Complex Var. Theory Appl. 37(1), 185–193 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gross, F.: Factorization of meromorphic functions and some open problems. In: Proc. Conf. Univ. Kentucky, Leixngton (1976)Google Scholar
  9. 9.
    Gross, F.: Lecture Notes in Mathematics, vol. 599, pp. 51–69. Springer, Berlin (1977)Google Scholar
  10. 10.
    Gross, F., Yang, C.C.: On preimage and range sets of meromorphic functions. Proc. Japan Acad. 58, 17–20 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hayman, W.K.: Meromorphic Functions. The Clarendon Press, Oxford (1964)MATHGoogle Scholar
  12. 12.
    Lahiri, I.: Value distribution of certain differential polynomials. Int. J. Math. Math. Sci. 28(2), 83–91 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lahiri, I.: Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161, 193–206 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lahiri, I.: Weighted value sharing and uniqueness of meromorphic functions. Complex Var. 46, 241–253 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, P., Yang, C.C.: Some further results on the unique range sets of meromorphic functions. Kodai Math. J. 18, 437–450 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mokhon’ko, A.Z.: On the Nevanlinna characteristics of some meromorphic functions. In: Theory of Functions, Functional Analysis and Their Applications. Izd-vo Khar’kovsk Un-ta, vol. 14, pp. 83–87 (1971)Google Scholar
  17. 17.
    Yi, H.X.: Unicity theorems for meromorphic or entire functions III. Bull. Austral. Math. Soc. 53, 71–82 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yi, H.X.: Meromorphic functions that share one or two values II. Kodai Math. J. 22, 264–272 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yi, H.X., Lü, W.R.: Meromorphic functions that share two sets, II. Acta Math. Sci. Ser. B (Engl. Ed.) 24(1), 8390 (2004)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KalyaniKalyaniIndia

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