zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Uniqueness theorems in an angular domain. (English) Zbl 1134.30026
Summary: There are many papers on the uniqueness theory of meromorphic functions in the whole plane $\bbfC$. However, the uniqueness theory concerned with shared sets in an angular domain does not yet seem widely investigated. In this paper, we deal with the problem of uniqueness for meromorphic functions in $\bbfC$ under some conditions in an angular domain instead of the whole plane. Moreover, examples show that those conditions are necessary.

30D55H (sup p)-classes (MSC2000)
30D35Distribution of values (one complex variable); Nevanlinna theory
Full Text: DOI
[1] A. Baernstein, Proof of Edrei’s spread conjecture, Proc. London Math. Soc. (3) 26 (1973), 418--434. · Zbl 0263.30024 · doi:10.1112/plms/s3-26.3.418
[2] J. Dufresnoy, Sur les fonctions méromorphes dans un angle, C. R. Acad. Sci. 208 (1939), 718--720. · Zbl 0020.23701
[3] A. Edrei, Sums of deficiencies of meromorphic functions, J. Analyse Math. 14 (1965), 79--107. · Zbl 0154.07402 · doi:10.1007/BF02806380
[4] A. Eremenko, I. V. Ostrovskii and M. Sodin, Anatolii Asirovich Gol’dberg, Complex Variables Theory Appl. 37 (1998), 1--51. · Zbl 1054.01007
[5] M. L. Fang and W. S. Xu, On the uniqueness of entire functions, Bull. Malaysian Math. Soc. (2) 19 (1996), 29--37. · Zbl 0880.30026
[6] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Variables Theory Appl. 37 (1998), 185--193. · Zbl 1054.30519
[7] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122 (2000), 1175--1203. · Zbl 0983.30013 · doi:10.1353/ajm.2000.0045 · http://muse.jhu.edu/journals/american_journal_of_mathematics/toc/ajm122.6.html
[8] A. A. Gol’dberg, Nevanlinna’s lemma on the logarithmic derivative of a meromorphic function, Mat Zametki 17 (1975), 525--529 (in Russian); Engl. transl.; Math. Notes 17 (1975), 310--312. · Zbl 0316.30021 · doi:10.1007/BF01105380
[9] A. A. Gol’dberg and I. V. Ostrovskii, The distribution of values of meromorphic functions, Izdat. Nauka, Moscow, 1970 (in Russian).
[10] F. Gross, Factorization of meromorphic functions and some problems, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky, 1976), 51--67, Lecture Notes in Math. 599, Springer-Verlag, Berlin, 1977. · Zbl 0357.30007
[11] W. K. Hayman, Meromorphic functions, Oxford Math. Monogr., Clarendon, Oxford, 1964. · Zbl 0115.06203
[12] I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Stud. Math. 15, Walter de Gruyter & Co., Berlin, 1993. · Zbl 0784.30002
[13] P. Li and C. C. Yang, On the unique range set of meromorphic functions, Proc. Amer. Math. Soc. 124 (1996), 177--185. JSTOR: · Zbl 0845.30018 · doi:10.1090/S0002-9939-96-03045-6 · http://links.jstor.org/sici?sici=0002-9939%28199601%29124%3A1%3C177%3AOTURSO%3E2.0.CO%3B2-Z&origin=euclid
[14] L. Liao and C. C. Yang, On the cardinality of the unique range set for meromorphic and entire functions, Indian J. Pure Appl. Math. 31 (2000), 431--440. · Zbl 0979.30019
[15] E. Mues and M. Reinders, Meromorphic functions sharing one value and unique range sets, Kodai Math. J. 18 (1995), 515--522. · Zbl 0919.30023 · doi:10.2996/kmj/1138043489
[16] R. Nevanlinna, Uber die Eigenschaften meromorpher Funktionen in einem Winkelraum, Acta Soc. Sci. Fenn. 50 (12) (1925), 1--45. · Zbl 51.0257.02
[17] I. V. Ostrovskii, The connection between the growth of meromorphic function and the distribution of the arguments of its values, Izv. Akad. Nauk. SSSR Ser. Mat. 25 (1961), 277--328.
[18] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. · Zbl 0087.28401
[19] C. C. Yang and X. H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395--406. · Zbl 0890.30019 · emis:journals/AASF/Vol22/vol22.html · eudml:228963
[20] L. Yang, Borel direction of meromorphic functions in an angular domain, Sci. Sinica 1979, Special Issue I on Math., 149--164.
[21] L. Z. Yang, Meromorphic functions that share two values, J. Math. Anal. Appl. 209 (1997), 542--550. · Zbl 0941.30021 · doi:10.1006/jmaa.1997.5329
[22] H. X. Yi, Uniqueness of meromorphic functions and a question of Gross, Science China Ser. A 37 (1994), 802--813. · Zbl 0821.30024
[23] H. X. Yi, Unicity theorems for meromorphic or entire functions, Bull. Austral. Math. Soc. 49 (1994), 257--265. · Zbl 0809.30024 · doi:10.1017/S0004972700016324
[24] H. X. Yi, Meromorphic functions that share one or two values, Complex Variables Theory Appl. 28 (1995), 1--11. · Zbl 0841.30027
[25] H. X. Yi, Unicity theorems for meromorphic or entire functions II, Bull. Austral. Math. Soc. 52 (1995), 215--224. · Zbl 0844.30022 · doi:10.1017/S0004972700014635
[26] H. X. Yi, Unicity theorems for meromorphic or entire functions III, Bull. Austral. Math. Soc. 53 (1996), 71--82. · Zbl 0855.30025 · doi:10.1017/S0004972700016737
[27] H. X. Yi, On a question of Gross concerning uniqueness of entire functions, Bull. Austral. Math. Soc. 57 (1998), 343--349. · Zbl 0905.30026 · doi:10.1017/S0004972700031701
[28] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Pure and Applied Math. Monographs 32, Science Press, Beijing, 1995.
[29] H. X. Yi and W. Lin, Uniqueness theorems concerning a question of Gross, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), 136--140. · Zbl 1112.30028 · doi:10.3792/pjaa.80.136 · euclid:pja/1116442331
[30] J. H. Zheng, On transcendental meromorphic functions with radially distributed values, Sci. China Ser. A 47 (2004), 401--416. · Zbl 1081.30032 · doi:10.1360/02ys0210
[31] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad. Math. Bull. 47 (2004), 152--160. · Zbl 1045.30019 · doi:10.4153/CMB-2004-016-1
[32] J. H. Zheng, On uniqueness of meromorphic functions with four shared values in one angular domain, Complex Var. Theory Appl. 48 (2003), 777--786. · Zbl 1041.30009 · doi:10.1080/02781070310001599368