UNIQUENESS RESULTS OF MEROMORPHIC FUNCTIONS WHOSE DERIVATIVES SHARE FOUR SMALL FUNCTIONS

doi:10.1016/S0252-9602(12)60126-X

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (4): 1593-1606.doi: 10.1016/S0252-9602(12)60126-X

• Articles • Previous Articles Next Articles

UNIQUENESS RESULTS OF MEROMORPHIC FUNCTIONS WHOSE DERIVATIVES SHARE FOUR SMALL FUNCTIONS

LI Xiao-Min^1,2, YI Hong-Xun¹, HU Hai-Yan³

1. Department of Mathematics, Ocean University of China, Qingdao 266100, China；
2. Department of Physics and Mathematics, University of Eastern Finland, P. O. Box 111, 80101 Joensuu, Finland；
3. Department of Mathematics, Shandong University, Jinan 250100, China

Received:2010-04-26 Revised:2011-07-28 Online:2012-07-20 Published:2012-07-20
Supported by:
This work is supported by the NSFC (11171184), the NSFC (10771121), the NSFC & RFBR (Joint Project) (10911120056), the NSFC (40776006), the NSF of Shandong Province, China (Z2008A01), and the NSF of Shandong Province, China (ZR2009AM008).

Abstract

Abstract:

We prove an oscillation theorem of two meromorphic functions whose deriva-tives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10], and supplement results given by
Nevanlinna [9] and Gundersen [3, 4]. Some examples are provided to show that the results in this paper are best possible.

Key words: Nevanlinna theory, meromorphic function, shared value, uniqueness, order of growth

CLC Number:

30D35

LI Xiao-Min, YI Hong-Xun, HU Hai-Yan. UNIQUENESS RESULTS OF MEROMORPHIC FUNCTIONS WHOSE DERIVATIVES SHARE FOUR SMALL FUNCTIONS[J].Acta mathematica scientia,Series B, 2012, 32(4): 1593-1606.

Trendmd

References

[1] Adams W W, Straus E G. Non-archimedian analytic functions taking the same values at the same points. Illinois J Math, 1971, 15: 418–424

[2] Chuang C T. Sur la comparaison de la croissance d’une fonction m´eromorphe et celle de sa dérivée. Bull Sci Math, 1951, 75: 171–190

[3] Gundersen G G. Meromorphic functions that share that share three or four values. J London Math Soc, 1979, 20(2): 457–466

[4] Gundersen G G. Meromorphic functions that share four values. Trans Amer Math Soc, 1983, 277: 545-567. Correction: 1987, 304: 847–850

[5] Gundersen G G. Meromorphic functions that share three values IM and a fourth value CM. Complex Variables Theory Appl, 1992, 20: 99–106

[6] Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964

[7] Ishizaki K, Toda N. Unicity theorems for meromorphic functions sharing four small functions. Kodai Math J, 1998, 21: 350–371

[8] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin, New York: Walter de Gruyter, 1993

[9] Nevanlinna R. Einige Eindeutigkeitss¨atze in der Theorie der meromorphen Funktionen. Acta Math, 1926, 48: 367–391

[10] Qiu G D. Four values theorem of derivatives of meromorphic functions. Sys Sci Math Sci, 1998, 11: 245–248

[11] Qiu G D. Meromorphic functions whose derivatives share four small functions. Indian J Pure Appl Math, 2002, 33: 761–767

[12] Steinmetz N. A uniqueness theorem for three meromorphic functions. Ann Acad Sci Fenn Ser A I Math, 1988, 13: 93–110

[13] Yamanoi K. The second main theorem for small functions and related problems. Acta Math, 2004, 192: 225–299

[14] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. Dordrecht, Boston, London: Kluwer Academic Publishers, 2003

[15] Yang L. Value Distribution Theory. Berlin, Heidelberg: Springer-Verlag, 1993

[16] Yang L. Precise estimate of total deficiency of meromorphic derivatives. J D'Anal Math, 1990, 55: 287–296

UNIQUENESS RESULTS OF MEROMORPHIC FUNCTIONS WHOSE DERIVATIVES SHARE FOUR SMALL FUNCTIONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Shinji ADACHI, Masataka SHIBATA, Tatsuya WATANABE. A NOTE ON THE UNIQUENESS AND THE NON-DEGENERACY OF POSITIVE RADIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS AND ITS APPLICATION [J]. Acta mathematica scientia,Series B, 2018, 38(4): 1121-1142.
[2]	Huifang LIU, Zhiqiang MAO. ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS [J]. Acta mathematica scientia,Series B, 2018, 38(3): 819-828.
[3]	Olivier GUIBÉ, Alip OROPEZA. RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS [J]. Acta mathematica scientia,Series B, 2017, 37(4): 889-910.
[4]	Nguyen Van THIN. NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE [J]. Acta mathematica scientia,Series B, 2017, 37(3): 623-656.
[5]	Ning CUI, Zong-Xuan CHEN. ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS [J]. Acta mathematica scientia,Series B, 2017, 37(3): 786-798.
[6]	Lingyun GAO. ON ENTIRE SOLUTIONS OF TWO TYPES OF SYSTEMS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS [J]. Acta mathematica scientia,Series B, 2017, 37(1): 187-194.
[7]	Zhendong LUO. A STABILIZED MIXED FINITE ELEMENT FORMULATION FOR THE NON-STATIONARY INCOMPRESSIBLE BOUSSINESQ EQUATIONS [J]. Acta mathematica scientia,Series B, 2016, 36(2): 385-393.
[8]	Wei CHEN, Honggen TIAN, Peichu HU. NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS WITH SHARED VALUES [J]. Acta mathematica scientia,Series B, 2016, 36(1): 87-93.
[9]	Sahbi BOUSSANDEL. EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR GRADIENT SYSTEMS IN FINITE DIMENSIONAL SPACES [J]. Acta mathematica scientia,Series B, 2016, 36(1): 233-243.
[10]	Yang TAN. SEVERAL UNIQUENESS THEOREMS OF ALGEBROID FUNCTIONS ON ANNULI [J]. Acta mathematica scientia,Series B, 2016, 36(1): 295-316.
[11]	Wenjun YUAN, Weiling XIONG, Jianming LIN, Yonghong WU. ALL MEROMORPHIC SOLUTIONS OF AN AUXILIARY ORDINARY DIFFERENTIAL EQUATION AND ITS APPLICATIONS [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1241-1250.
[12]	Zhaojun WU, Sheng'an CHEN, Yuxian CHEN. CHARACTERISTIC FUNCTION AND DEFICIENCY OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED PLANE [J]. Acta mathematica scientia,Series B, 2015, 35(3): 673-680.
[13]	Fang LIU. SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS [J]. Acta mathematica scientia,Series B, 2015, 35(2): 477-494.
[14]	LI Hai Chou,GAO Ling Yun. MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS [J]. Acta mathematica scientia,Series B, 2015, 35(1): 195-206.
[15]	CAO Hong-Zhe, CAO Ting-Bin. TWO MEROMORPHIC FUNCTIONS SHARE SOME PAIRS OF SMALL FUNCTIONS WITH TRUNCATED MULTIPLICITIES [J]. Acta mathematica scientia,Series B, 2014, 34(6): 1854-1864.