Taiwanese Journal of Mathematics

Unique Range Sets for Meromorphic Functions Constructed without an Injectivity Hypothesis

Ta Thi Hoai An

Full-text: Open access


A set is called a unique range set (counting multiplicities) for a particular family of functions if the inverse image of the set counting multiplicities uniquely determines the function in the family. So far, almost all constructions of unique range sets for meromorphic functions are zero sets of polynomials which satisfy an injectivity condition introduced by Fujimoto. A polynomial $P(z)$ satisfies the injectivity condition if $P$ is injective on the zeros of its derivative. In this paper, we will construct examples of unique range sets for meromorphic functions without assuming an injectivity condition.

Article information

Taiwanese J. Math., Volume 15, Number 2 (2011), 697-709.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D30: Meromorphic functions, general theory 30D35: Distribution of values, Nevanlinna theory

unique range set uniqueness polynomial functional equation


An, Ta Thi Hoai. Unique Range Sets for Meromorphic Functions Constructed without an Injectivity Hypothesis. Taiwanese J. Math. 15 (2011), no. 2, 697--709. doi:10.11650/twjm/1500406229. https://projecteuclid.org/euclid.twjm/1500406229

Export citation


  • Ta Thi Hoai An, A new class of unique range sets for meromorphic functions on $\mathbb{C}$. Dedicated to the memory of Le Van Thiem (Hanoi, 1998), Acta Math. Vietnam., 27(3) (2002), 251-256.
  • Ta Thi Hoai An and Julie T.-Y Wang, Uniqueness polynomials for complex meromorphic functions, Internat. J. Math., 13(10) (2002), 1095-1115.
  • Ta Thi Hoai An, Julie T.-Y Wang and Pit-Mann Wong, Strong uniqueness polynomials: the complex case, Journal of Complex Variables and it's Application, 49(1) (2004), 25-54.
  • William Cherry and Julie T-Y Wang, Uniqueness polynomials for entire functions, Internat. J. Math., 13(3) (2002), 323-332.
  • Ming-Liang Fang and Indrajit Lahiri, Unique range set for certain meromorphic functions, Indian J. Math., 45(2) (2003), 141-150.
  • Günter Frank and Martin Reinders, A unique range set for meromorphic functions with $11$ elements, Complex Variables Theory Appl., 37(1-4) (1998), 185-193.
  • Hirotaka Fujimoto, On uniqueness polynomials for meromorphic functions, Nagoya Math. J., 170 (2003), 33-46.
  • Hirotaka Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122(6) (2000), 1175-1203.
  • Fred Gross and Chung Chun Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 58(1) (1982), 17-20.
  • Ha Huy Khoai, Some remarks on the genericity of unique range sets for meromorphic functions, Sci. China Ser. A, 48 (2005), suppl., 262-267.
  • Ping Li and Chung-Chun Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18(3) (1995), 437-450.
  • Feng Lü and Jun-feng Xu, Some further results on the unique range sets of meromorphic functions, Northeast. Math. J., 23(4) (2007), 335-343.
  • Gary G. Gundersen and Kazuya Tohge, Unique range sets for polynomials or rational functions, Progress in analysis, Vol. I, II (Berlin, 2001), 235-246, World Sci. Publ., River Edge, NJ, 2003.
  • Yum-Tong Siu and Sai-Kee Yeung, Defects for ample divisors of Abelian varieties, Schwarz lemma, and Hyperbolic hypersurfaces of low degrees, Amer. J. Math., 119 (1997), 1139-1172.