×

zbMATH — the first resource for mathematics

Unicity theorems for meromorphic functions that share three values. (English) Zbl 0868.30032
Two meromorphic functions \(f\) and \(g\) share the complex value \(a\) if \(f(z)=a\) implies \(g(z)=a\) and vice versa. The value \(a\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(a\)-point.
Let \(f\) and \(g\) be distinct non-constant meromorphic functions in the complex plane sharing the value \(0\), \(1\) and \(\infty\) CM.
The main result of the paper under review is Theorem 4: If \[ N(r,1/(f-a))\neq T(r,f)+ S(r,f)\tag{\(*\)} \] then \(f\) and \(g\) are related by one of the following relations: \((f-a)(g+a-1)= a(1-a)\), \(f+(a-1)g=a\) or \(f=ag\).
G. Brosch [Eindeutigkeitssätze für meromorphe Funktionen, Dissertation, RWTH Aachen (1989; Zbl 0694.30027)] already proved (Folgerung 5.2, which is a corollary to a result of E. Mues) that the condition \((*)\) implies \(f=L(g)\) with a Möbius transformation \(L\). The conclusion of Theorem 4 then follows immediately. The proof in the present paper is almost identical to the proof of G. Brosch.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
30D20 Entire functions of one complex variable (general theory)
Keywords:
sharing values
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. G. GUNDERSEN, Meromorphic functions that share three or four values, J. London Math. Soc, 20 (1979), 457-466. · Zbl 0413.30025
[2] W. K. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964 · Zbl 0115.06203
[3] H. UEDA, Unicity theorems for meromorphic orentire functions, Kodai Math. J., 3 (1980), 457-471 · Zbl 0468.30023
[4] HONG-XUN Yi, Meromorphic functions that share three values, Chinese Ann Math., 9A (1988), 434-439. · Zbl 0699.30024
[5] SHOU-ZHEN YE, Uniqueness of meromorphic functions that share three values, Kodai Math. J., 15 (1992), 236-243 · Zbl 0767.30026
[6] HONG-XUN Yi, Meromorphic functions that share two or three values, Koda Math. J., 13 (1990), 363-372. · Zbl 0712.30029
[7] HONG-XUN YI, Unicity theorems for meromorphic functions, J. Shandong Univ., 23 (1988), 15-22 · Zbl 0702.30029
[8] G. BROSCH, Eindeutigkeitssatze fur meromorphc Funktionen, Thesis, Technica University of Aachen, 1989.
[9] F. GROSS, On the distribution of values of meromorphic functions, Trans. Amer Math. Soc, 131 (1968), 199-214. · Zbl 0157.12903
[10] HONG-XUN YI, On a result of Gross and Yang, Thoku Math. J., 42 (1990), 419-428 · Zbl 0714.30028
[11] F. GROSS AND C. C. YANG, Moromorphic functions covering certain finite set at the same points, Illinois J. Math., 26 (1982), 432-441. · Zbl 0503.30029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.