| Normal family theory is an important branch of complex analysis, which not only plays a significate role in basic research, but also has its own ap-plications in other branch. For instance, normal family combined with value distribution theory could yield many valuable research works, and as a part of complex dynamics leads to the definitions of Julia set and Fatou set. Normal family has been experiencing a noble development since P. Montel initiated it in the prime of 20th century. Especially, some scholars from China, such as Q. L. Xiong, Q. T. Zhuang, L. Yang, Y. X. Gu, M. L. Fang and X. C. Pang, have gained many excellent results in normal family, which brings China a leadership in this field. In Chapter 1, we will introduce the basic concepts, theorems, and some preliminary knowledge which is necessary to normal family theory, such as Nevanlinna value distribution.A key stage in the history of normal family theory came up in 1950s, and then lasted till 1980s. Among this stage, scholars mainly studied several conjectures posed by W. K. Hayman, and thus obtained a series of normal criteria. The main issue of our article comes from a conjecture in value distribution, which is just a question proposed by Hayman [17]. The con-jecture shows that If F is transcendental, then FnF' assumes every finite non-zero complex number infinitely often for any positive integer n. About this conjecture, Hayman ([17]), E. Mues [35], J. Clunie [8], W. Bergweiler and A.Eremenko [3], H. H. Chen and M. L. Fang [4] all got their theorems, and finally confirmed it.According to the Bloch's principle, Hayman [19] proposed another con-jecture related to above problem on value distribution, it as follows:If each f?F satisfies fnf'?a for a positive integer n and a finite non-zero com-plex number a, then F is normal in D. This conjecture has been shown to be true by L. Yang and Q. Zhuang [59] for n?5; Gu [14] for n=3,4, Oshkin [36] for holomorphic functions, n=1; (cf. [28])and then X. Pang [38] for n?2. in general; cf. [14]). As indicated by X. Pang [38], the conjecture for n=1 is a consequence of Chen-Fang's theorem (cf. [4], [65], [64]).Lately, Q. C. Zhang [67] proved that a family of meromorphic functions F is also normal in D when each pair (f,g) of F is such that fnf' and gng' share a finite non-zero complex number a IM for n?2. There are examples showing that this result is not true if n=1. Comparing Hayman's conjectures with Zhang's theorem, we could find a explicit way from value distribution to normal criteria of the Picard type, and then to criteria related to sharing values. Furthermore, for the case of high derivatives such as FnF(k),similar results were obtained by Q. C. Zhang and J. M. Qi, see [68], [43].To understand the problem about FnF(k) well, many authors studied the functions of the form F (F(k))n along the researching route of Hayman's problem. In 1993, C. C. Yang, L. Yang and Y. F. Wang [55] proved that if n?2 and F is a transcendental entire function, then the only possible Picard value of F (F(k))n is the value zero. In 1998, Z. F. Zhang and G. D. Song [69] announced that, if F is transcendental; a(?){0,?}:n?2, then F (F(k))n-a has infinitely many zeros. A simple proof was given by A. Alotaibi [1]. In fact, they proved a better result showing that this fact is true if a ((?)0) is a small meromorphic function of FIn Chapter 2, according to the Bloch's principle and also the results confirmed by C. Yang, L. Yang, Y. Wang, Z. Zhang and G. Song, and Alotaibi respectively, we mainly discuss the normality of F (F(k))n, and then obtained a new criterion as follows:(1) Take positive integers n and k with n,k?2 and take a non-zero complex number a. Let F be a family of meromorphic functions in the plane domain D such that each f?F has only zeros of multiplicity at least k. For each pair (f,g) of?F, if f(f(k))n and g(g(k))n share a IM, then F is normal in D.We additionally give some examples. Set D={z?C||z|<1}, then shows that the condition that f has only zeros of multiplicity at least k is sharp. Moreover, the condition k?2 is necessary. For example, the following family shows that the above theorem is not true for the case k=1.Additionally, based the above result and its proof, we got two corollaries as follows:(2) Take positive integers n and k with n?2 and take a non-zero complex number a. Let F be a family of meromorphic functions in the plane domain D such that each f?F has only zeros of multiplicity at least k. For each clement f of F, if f(z) (f(k)(z))n=a implies|f(k)(z)|?A for a positive number A, then F is normal in D.(3) Take positive integers n and k with n?2 and take a non-zero complex number a. Let F be a family of meromorphic functions in the plane domain D such that each f?F has only zeros of multiplicity at least k. For each element f of F satisfies f(z) (f(k)(z))n?/a for any z?D, then F is normal in D.In Chapter 2, all normal criteria require the condition n?2. It is natural to ask wether the criteria hold or not when n=1. L. Yang and C. C. Yang [58] proposed the conjecture:If F is transcendental, then FF(k) assumes every finite non-zero complex number infinitely often for any positive integer k. C.C. Yang and P. C. Hu [54] obtained a part of answer. In 2006, J. P, Wang [49, Theorem 3] proved that this conjecture holds when F has only zeros of multiplicity at least k+1 (K?2). Influenced by Bloch's principle, it is natural to ask wether normality criteria corresponding to FF(k) exit or not. In a conference (Shanghai,2009), M. L. Fang proposed a conjecture: Let a be a non-zero complex number,F a family of mcromorphic functions in D(?)C. If each f?F, we have ff(k)?a, then F is normal in D.In Chapter 3, we will discuss the normality of meromorphic or holo-morphic functions sharing one value, especially the form of FF(k),and thus obtain two normal criteria:(4) Take a positive integer k and a non-zero complex number a. Let F be a family of mcromorphic functions in a domain D(?)C such that each f?F has only zeros of multiplicity at least k+1. For each pair (f,g)?F, if ff(k) and gg(k) share a IM, then F is normal in D.(5) Take a positive integer k?2 and a non-zero complex number a. Let F be a family of holomorphic functions in the a domain D(?)C such that each f?F has only zeros of multiplicity at least k. For each pair (f,g)?F, if ff(k) and gg(k) share a IM, then F is normal in D.Added the condition that multiplicity at least k+1, theorem (4) ob-viously answered the unsolved problem of [21] for the case of n= 1, and partially solve Fang's conjecture.Finally, in Chapter 4, we mainly discuss the uniqueness of differential polynomials sharing three values such as 0,1,?. A main theorem is the following, which extends the results of I. Lahiri [23], Q. C. Zhang [66].(6) Let f,g be two nonconstant mcromorphic functions, and let?n(f),?n(g) be non-constant such that:(i)?n(f) and?n(g) share 0 IM,?IM,1 CM, Then cither (a)?n(f)?n(g)=1 or (b)f-g=q, where q=q(z) is a solution of the differential equation?n(q)=0.Obviously, in above theorem, if f has at least one pole or?n(f) has at least one zero, then the possibility (a) does not occur. |
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