| The value distribution theory was established by Rolf Nevanlinna in 1920's. In honor of him, we usually call it Nevanlinna theory. This theory can be seen as the best achievements during the last century to study the properties of mero-morphic functions. Nevanlinna theory contains two main theorems, namely the fist and the second fundamental theorems. The two main theorems can be widely used in many problems. For instance, the latter one greatly extends the Picard's first theorem. As the development of Nevanlinna theory, it has lots of applica-tions in many fields of mathematics, for example, uniqueness of meromorphic functions, normal family, complex differential equations and dynamics theories etc.Difference counterparts of Nevanlinna theory have been established very re-cently. The main tool is the difference analogue of the lemma on the logarithmic derivative, which was obtained by Halburd-Korhoncn [16] and Chiang-Feng [9], independently. Related to q-difference operator, Barnett etc. [2] got the corre-sponding logarithmic derivative lemma. Recently, Zhang and Korhonen [56] used the q difference analogue of the lemma on the logarithmic derivative to investi-gate the relationship between T(r, f(qz)) and T(r, f(z)), and they also gave some applications.In the early twentieth century, P. Montel gave the definition of normal family. Let F be a family of meromorphic functions in a domain D?C. We say that F is normal in D if every sequence{fn}?F contains a subsequence which converges spherically and uniformly on compact subsets of D (see [36]). Since then, with the development of Nevanlinna theory, normal family has also developed rapidly. Many mathematicians in China such like Q.L. Xiong, C.T. Chuang, L. Yang, G.H. Zhang etc. have made great contributions to the development of the theory. In recent decades, especially the appearance of Zalcman-Pang's Lemma, made the development of normal family theory enter a new world. During this period, many Chinese mathematicians, such as Y.X. Gu, H.H. Chen, X.C. Pang, M.L Fang, J.M. Chang, etc. have done a number of beautiful jobs and made great contributions to the development of the theory in the world.Under the guidance of my supervisor Prof. H.X. Yi, the present thesis con-tains some results of the author that consider the uniqueness of meromorphic functions or q-difference operator sharing a small function, and normality criteria for families of some functions concerning Lahiri's type and their applications on a problem related to R. Bruck's Conjecture. It includes four parts and the matters are explained as follows.In Chapter 1, we introduce the basic background of Nevanlinna theory, the development of the normal family theory, and a very important theory called Wiman-Valion theory, which is often used in the field of complex differential equations.In Chapter 2, we consider uniqueness of meromorphic functions sharing a small function. The uniqueness problem of meromorphic functions sharing one value is very interesting in uniqueness problems, a great number of good results in this directions have appeared (see e.g. [11,12,13,26,41,48,55,58,59,60]). Here we do further consideration and obtain a more general result as follows.Theorem 0.1. Let f and g be two non-constant meromorphic functions, a(z)((?) 0,?) be a small function with respect to f. Let n, k, and m be three positive in-tegers with n> 3k+n+8 and P(?) be defined as in Theorem G. If [fnP(f)]k and [gnP(g)](k) share a CM, then(I) when P(w)= amwm+am-1wm-1+...+ a1w+a0, one of the following three cases holds:(II) f(z)= tg(z) for a constant t such that td=1, where d=GCD(n+ m,...,n+m-i,
,n),am-it?0 for some i=0,1,
,m,(12)f and g satisfy the algebraic equation R(f,g)?Q,where R(?l,?2)?n1(am?m1?am-1?m-11+...+a0)-?n2(am?m2+am-1?m-12W+
+a0),(13)[fnP(f)](k)[gnp(g)](k)=a2;(?)when p(w)?c0,one of the following two cases holds:(?1)f?tg for a constant t such that tn=1,(?2)c20[fn](k)[gn](k)=a2.We also present some applications of Theorem 0.1,the following result we obtained has improved or generalized many previous results.Theorem 0.2.Let f and g be two nonconstant meromorphic functions,a(z)(?O,?)be a small function with respect to f with finitely many zeros and poles. Let n,k and m be three positive integers with n>3k+m+7,P(w)=amwm+ am-1wm-1+
+a1w+a0,where a0?0,a1,
,am-1,a,am?0,are complex constants.If[fnP(f)](k)and[gnp(g)](k) share a CM,and q share?IM,then one of the following two cases holds: (1) f(z)?tg(z)for a constantt such that td=1,where d=GCD(n+m,
,n+ m-i,
,n),am-i?0 for some i=0,1,
,m; (2) f and g satisfy the algebraic equation R(f,g)?0,where R(?1,?2)=?n1(am?m1+ am-1?m-11+
+a0)-?n2(am?m2+am-1?m-12+
+a0).Theorem 0.3.Let f and g be two transcendental meromorphic functions,p(z) be a nonzero polynomial with deg(p)=l?5,n,k and m be three positive inte-gers with n>3k+m+7.Let P(?)=am?m7+am-1?m-1+
+a1?+a0 be a nonzero polynomial.If[fnp(f)](k)and [gnp(g)](k))share p CM,and g share?IM.Then one of the following three cases holds:(1) f(z)?tg(z)for a constant t such that td=1,where d=GCD(n+m,
,n+ m-i,
,n),am-i?0 for some i=0,1,
,m;(2) f and g satisfy the algebraic equation R(f,9)?0,where R(?1,?2)=?n1(am?m1+ am-14?m-11+
+a0)-?n2(am?m2+am-1?m-12+
+a0);(3) p(z)is reduced to a nonzero monomial, namely,P(z)=aizi?0 for some i?{0,1,...,m};if p(z)is not a constant,then f=c1eCQ(z),g=c2e-cQ(z),where Q(z)=?z0p(z)dz,c1,c2 and c are constants such that a2i(c1c2)n+i[(n+i)c]2=-1, if p(z) is a nonzero constant b,then f=c3ecz,g=c4e-cz,where c3,c4 and c are constants such that(?1)ka2i(c3c4)n+i+[(n+i)c2k]=b2.In Chapter 3,we consider the uniqueness and value distribution of q-difference functions.Related to the rosearch on such directions,recently, Zhang and Ko-rhonen[56] considered the case of entire functions.We generalize their results to the case of meromorphic functions.We obtainTheorem 0.4. Let f(z)and g(z)be two non-constant meromorphic functions of zero-order. Suppose that q is a nonzero complex constant and n?14 is an integer.If fn(z)f(qz)andgn(z)g(qz) share 1 CM,f and g have at least one common pole,then f(z)?tg(z),where t is a constant such that tn+1=1.Theorem 0.5.Let f(z) and g(z) be two transcendental meromorphic functions of zero-order. Suppose that q is a nonzero complex constant and n?14 is an integer.If fn(z)f(qz)and gn(z)g(qz) share z CM,then f(z)?tg(z)fortn+1=1.Theorem 0.6.Let f(z)and g(z)be two non-constant meromorphic functions of zero-order.Suppose that q is a constant(|q|?0,1) and n?15 is an integer. If fn(z)(f(z)?1)f(qz)and gn(z)(g(z)?1)g(qz)share 1 CM,(z)and g(z)share?IM, then fn(z)(f(z)?1)f(qz)?gn(z)(g(z)?1)g(qz).We have also considered the value distribution of zeros of fn(z)+a(z)f(qz)-b(z),which can be seen as a q-difference analogue of fn+af'-b,see Hayman [19].We getTheorem 0.7.Let f be a transcendental meromorphic function of zeroorder, and a(z)(?0,?),b(z)(??)be small functions of f(z).Then fn(z)+a(z)f(qz)-b(z) has infinitely many zeros forn?6. If f(z)is transcendental entire, this holds for n?2.Moreover,we obtain a uniqueness theorem corresponding to Theorem 0.7.Theorem 0.8.Let f(z)and g(z) be two trabscendetal entire functions of zero-order, n?7 be an integer,q be a nonzero constant,and a(z)(?0,?),b(z)(?0,?) be small functions with respect to both f(z)and g(z).If fn(z)+a(z)f(qz) and gn(z)+a(z)g(qz)share b(z)CM,then f(z)?g(z).We also gave some examples to show the requircment of some conditions in Theorems 0.4-0.7. In chapter 4, we prove two normality criteria for families of some functions concerning Lahiri's type, the results generalize those given by Charak and Rieppo [6], Xu and Cao [40]. The results we obtained are as follows.Theorem 0.9. Let F be a family of meromorphic functions in a complex domain D, for every f?F, all zeros of f have multiplicity at least k. Let a, b?C such that a?O, let m, n, k(?1), mj, nj (j= 1,2,...,k) be nonnegative integers such that?M2?M1-?M1?m2> 0, nk+mk> 0, m+n?2. Put If there exists a positive constant M such that|f(z)|?M for all f?F whenever z?Ef, then F is a normal family.Theorem 0.10. Let F be a family of meromorphic functions in a complex domain D, for every f?F, all zeros of f have multiplicity at least k. Let a, b?C such that a?0, let m, n, k(?1), mj, nj (j= 1,2,...,k) be nonnegative integers such that mnmknk?*m1*m2> 0, (k?2 when n=1 orm=1), m/n=mj/nj for all positive integers mj and nj, (1?j?k). Put If there exists a positive constant M such that|f(z)|> M for all f?F whenever z?Ef, then F is a normal family.As an application of Theorem 0.9, we obtain the following theorem.Theorem 0.11. Let F be a family of holomorphic functions in a domain D, for every f?F, all zeros of f have multiplicity at least k. Let a, b(?0) be two finite values and n,k,n1,...,nk be nonnegative integers with n?1, k?1, nk?1. For every f?F, all zeros of f have multiplicity at least k, if P(f)=a<=> M1(f,f',...,f(k))=b, then F is normal in D.Further more, using Theorem 0.11, we study a problem related to R. Briick's Conjecture and obtain a result that generalizes those given by Yang and Zhang [53], Lii, Xu and Chen [29]. We state the result as follows. Theorem 0.12. Let n,k,n1,...,nk be nonnegative integers with n?1,k?1,nk?1,and a,b be two finite nonzero values. Let f be a non-constant en-tire function whose zeros have multiplicity at least k. If fn+n1+...+nk=a(?) fn(f')n1...(f(k))nk=b,then where c is a nonzero constant. Specially, if a=b, then f=c1ewz,where c1 is a nonzero constant,?is the root of t(?)M1=1... |
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