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Some further results on the unique range sets of meromorphic functions. (English) Zbl 0849.30025
Let $S$ be a subset of the complex plane $\bbfC$. For a non-constant meromorphic function $f$ let $E_f(S)= \bigcup_{n\in S} \{(z, p)\in \bbfC\times N\mid f(z)= a$ with multiplicity $p\}$. $S$ is called a unique range set for entire functions (URSE) if for any pair of non-constant entire functions $f$ and $g$ the condition $E_f(S)= E_g(S)$ implies $f= g$. Unique range sets for meromorphic functions (URSM) are defined in a similar way. The existence of URSE with finitely many elements has been exhibited by {\it H.-X. Yi} [Sci. China, Ser. A 38, No. 1, 8-16 (1995; Zbl 0819.30017)], thereby solving a question of {\it F. Gross} [Complex analysis, Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51-67 (1977; Zbl 0357.30007)]. In the paper under review, the authors show that there exist URSE with 7 elements and URSM with 15 elements. The proof uses Nevanlinna’s theorem on the so-called “Borel-identities” and a very careful analysis of the zeros and poles of the Wronskian determinant occurring in this theorem. Lower bounds for the cardinality of unique range sets are also given. To determine the exact value of the minimum cardinality of unique range sets is an interesting but still unsolved problem in value distribution theory.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
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References:
[1] F. GROSS, Factorization of meromorphic functions and some open problems, Com-plex Analysis, Lecture Notes in Math., 599, Springer, 1977, 51-67. · Zbl 0357.30007
[2] F. GROSS AND C. C. YANG, On preimage and range sets of meromorphic functions, Proc. Japan Acad., 58 (1982), 17-20 · Zbl 0501.30026 · doi:10.3792/pjaa.58.17
[3] F. GROSS, Factorizations of Meromorphic Functions, U. S. Govt. Printing Offic Publication, Washington D. C., 1972. · Zbl 0266.30006
[4] G. G. GUNDERSEN AND C. C. YANG, On the preimage sets of entire functions, Kodai Math. J., 7 (1984), 76-85 · Zbl 0548.30024 · doi:10.2996/kmj/1138036858
[5] W. K. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964 · Zbl 0115.06203
[6] R. NEVANLINNA, Le Theoreme de Picard-Borel et la Theorie des Functions Mero morphes, Gauthier-Villars, Paris, 1929.
[7] NONG LI AND GUODONG SONG, On a Gross’ problem concerning unicity of mero morphic functions, · Zbl 0889.30028
[8] PING LI AND C. C. YANG, On the unique range set of meromorphic functions, to appear in Proc. Amer. Math. Soc · Zbl 0845.30018 · doi:10.1090/S0002-9939-96-03045-6
[9] C. C. YANG, On deficiencies of differential polynomials, II, Math. Z., 125 (1972), 107-112 · Zbl 0217.38402 · doi:10.1007/BF01110921 · eudml:171694
[10] HONGXUN Yi, On a problem of Gross, Sci. China Ser. A, 24 (1994), 1137-1144