Uniqueness Of Meromorphic Functions With Multiple Zeros And Multiple Poles

This paper will be divided into three chapters.The first chapter introduces theknowledge which is prepared for the theory of value distribution, and the remaining twochapters are introduced the uniqueness of the function.The second chapter is to introduce the uniqueness of meromorphic function, whichconsiders the number of poles and zeros of the two meromorphic function.It improvedthe related results of S.S.Bhoosnurmath and R.S.Dyavanal and got a diffuse uniquenessresults.Theorem1Let f（z）, and g（z） be two meromorphic functions, whose zeros and polesare of multiplicities at least s,where s is a positive integer.Let n,k be two positiveintegers such that n2kand ns>3k+8.If [fⁿ]^{（k）} and [gⁿ]^{（k）} share1CM,theneither f（z）=c₁e^cz and g（z）=c₂e^-cz,or f=tg for tⁿ=1wherec, c₁and c₂aresatisfying （-1）^k（c₁c₂）ⁿ （nc）^2k=1.In the same way,we improve the results of Li Jindong and Lu Qian.Theorem2Let f（z） and g（z） be two non-constant meromorphic functions,whosezeros and poles are of multiplicities at least s,where s is a positive integer.Let n,k be twopositive integers such that n>2k and ns>6k+14,If [fⁿ]^{（k）} and [gⁿ]^{（k）}IM, theneither f（z）=c₁e^cz and g（z）=c₂e^-（cz）,wherec, c₁and c₂are three constants satisfying（-1）^k （c₁c₂）ⁿ （nc）^2k=1,or f=tg for a constant t such that tⁿ=1.Theorem3Let f（z） and g（z） be two non-constant meromorphic functions,whosezeros and poles are of multiplicities at least s,where s is a positive integer.Let n,k be twopositive integers such that n>4k and ns>6k+20,If [fⁿ （f-1）]^{（k）} and [gⁿ （g-1）]^{（k）} share1IM,then f（z）=g（z）.The third chapter is about the uniqueness of the polynomial with a weightedsharing1of entire function, and improves the results of Lin Xiuqing and Lin Weichuan.Theorem4Let f（z） and g（z） be two non-constant entire functions, n, m, k be threepositive integers such that n>k+（4k+7）（1-Θ（0; f,g））+4m （1-δ （1; f, g）），p≥2is aninteger,ifE_P)(1,（fⁿ （f-1）^m）^{（k）})=E_P)(1,（gⁿ （g-1）^m）^{（k）})，then either f （z）=g （z） or f,gsatisfy the algebraic equation R （f, g）0,whereR （w₁, w₂） w₁ⁿ （w₁-1）^m-w₂ⁿ （w₂-1）^m.