Uniqueness Of Meromorphic Functions With Their Differential And Q-difference Polynomials

Value distribution theory of meromorphic functions[10,30], was due to R.Nevanlinna in 1920's, is one of the most important achievements in mathematics in the preceding century. Its main contents consist of two main theorems, which are called Nevan-linna's first and second theorems. Since then, Nevanlinna theory has been well developed in itself and widely applied to the researches of the uniqueness of mero-morphic functions, differential and difference equations, normal families and Several Complex Variables etc.In 1929, R.Nevanlinna applied the value distribution theory[10,41] to consider the conditions under which a meromorphic function could be determined and de-rived the famous Nevanlinna's five-value and four-value theorems. From then on, many abroad and domestic mathematicians have devoted themselves to the researh and obtained lots of elegant results on the uniqueness theory.Difference counterpart of the theory of Nevanlinna theory have been estab-lished very recently. The key result is the difference analogue of the lemma on the logarithmic derivative obtained by Halburd-Korhonen [28]and Chiang-Feng [25]in-dependently. Halburd-Korhonen [26] also established a version of Nevanlinna theory for difference operators. Ishizaki and Yanagihara [31]also developed a version of Wiman-Valiron theory for slowly growing entire solutions of difference equations. Bergweiler and Langley [24]considered the value distributions of difference; opera- tors of slowly growing meromorphic functions.The present thesis involves some results of the author that investigate the uniqueness of meromorphic functions with their differential and q-difference poly-nomials,under the guidance of superwisor professor Lianzhong Yang. It has three chapters.In chapter 1,we introduce the background of this thesis,Nevanlinna basic the-ory and some fundamental results and notations of the uniqueness which would be used in the next two chapters.In chapter 2, we studied the uniqueness of meromorphic function with their differential polynomials and gained the following theorem.Theorem 1.Let f be a non-constant meromorphie function and Q[f] be a non-constant differential polynomial of degree d and weightГ.Let a(z)be a small meromorphic function of f such that a(z)≠0,∞.Suppose that f→a and Q[f]-a share(O,l),and(n-1)d≤(?)dMj.Then(?)=C for some non-zero constant C if one of the following assumptions holds,(1)l≥2 and 2N(r,f)+N2(r,1/Q)+N2(r,(?))<(λ+o(1))T(r,Q),(2)l=1 and 2N(r,f)+N2(r,1/Q)+2N(r,(?))<(λ+o(1))T(r,Q),(3)l=0 and 4N(r,f)+3N2(r,1/Q)+2N(r,(?))<(λ+o(1))T(r,Q), for r∈I,where 0<λ<1 and I is a set of infinite linear measure.In Chapter 3,we studied the uniqueness of meromorphic functions with their q-difference polynomials.Main results are stated as follows.Theorem 2. Let f(z) be a transcendental entire function of zero order,and a(z)∈S(r,f).Suppose that q is a non-zero complex constant and n is an integer.If m(P)>0,then P(f)f(qz)-a(z) has infinitely many zeros on a set of logarithmie density 1.Theorem 3.Let f(z) and g(z) be transcendental entire functions of zero order, a(z)be a small function with respect to both f(z)and g(z), and q be a non-zero coInstant.If n≥5s(P)+7m(P)+5,s(P)+m(P)≥2,P(f)f(qz) and P(g)g(qz) share a(z) IM,then on a set of logarithmic density 1,one of the following holds:(1)f≡tg for a constant t such that td=1,d=(i+1,…,k,…,n+1) (ak≠0),and ai be the first nonzero coefficient from the right.(2) f and g satisfy the algebraic equation P(f)f(qz)-P(g)g(gz)≡0.Corollary 1. Let f(z) and g(z) be transcendental entire functions of zero order,a(z) be a small function with respect to both f(z) and g(z),and q be a non-zero constant.If n≥4m+12,f(z)n(f(z)m-1)f(qz) and g(z)n(g(z)m-1)9(qz) share a(z)IM,then f≡tg on a set of logarithmic density 1,where tm=tn+1=1.Theorem 4. Let f be a non-constant meromorphic function of zero order, |q|(>1)be a non-zero complex constant,and let F=P(f).Suppose that F and F(qz) share a(z)∈S(r,f)\{O} and∞CM.If n≥3(s(P)+m(P))+1,then one of the following holds:(1)f(qz)≡ωf(z)for a constantωsuch thatωd=1,d=(i+1,...,k,...,n+1) (ak≠0),and ai be the first nonzero coefficient from the right.(2)f(qz)and f(z) satisfy the algebraic equation P(f(qz))≡P(f(z)).Corollary 2.Let f be a non-constant entire function of zero order,|q|(>1)be a non-zero complex constant,,and let F=P(f).Suppose that F and F(qz)share a(z)∈S(r,f)\{O} and∞CM.If n≥2(s(P)+m(P))+1,then one of the following holds:(1)f(qz)≡ωf(z)for a constantωsuch thatωd=1,d=(i+1,…,k,...,n+1) (ak≠0),and ai be the first nozero coefficient from the right.(2)f(qz)and f(z)satisfy the algebraic equation P(f(qz))≡P(f(z)).