| Value distribution theory of meromorphic functions, created by R. Nevanlinna in 20's, and in geometric form by L. Ahlfors about a decade later, is one of the most important achievements in the preceding century to understand the properties of meromorphic functions. Moreover, value distribution theory and its extensive have found a number of applications in other related fields of mathematics such as potential theory, several complex variables, complex differential, difference and functional equations, minimal surfaces etc.The uniqueness theory of meromorphic functions, essentially developed along with the Nevanlinna theory, is devoted to studying conditions that are satisfied by a few meromorphic functions only, or even determine a meromorphic function uniquely. The first results of this type within the value distribution theory were due to R. Nevanlinna [39]. These results are usually called Nevanllina's five-value, resp. four-value, theorem, meaning that whenever two meromorphic functions take five, resp. four, extended complex values at the same points in the complex plane, these two functions actually agree, resp. are Mobius transformations of each other. These two theorems are the starting points of the uniqueness theory, essentially developed during the last four decades, being presently an extensive theory, see, e.g., [44]. In addition to Nevanlinna type considerations comparing two meromorphic functions, attention has been also directed to uniqueness studies of a meromorphic function and its derivatives, resp. a meromorphic function and its differential polynomials, see [44], Chapter 8, and most recently to a meromorphic function and its difference polynomials, see, e.g. [22].For a Fermat type functional equationf(z)n+g(z)n + h(z)n = 1, (0.0.1)it is well known that there exist nonconstant meromorphic solutions satisfying (0.0.1) when n = 2,3,4. For the cases n≥9, W. K. Hayman[21] proved that there do not exist distinct transcendental meromorphic functions f,g and h that satisfy (0.0.1). Recently G. G. Gun-dersen considered the cases n = 5 and n = 6 [14, 15], and verified the existence of distinct transcendental meromorphic solutions of the equation (0.0.1) by his examples. To our best knowledge, the cases n = 7 and n = 8 are still open.This dissertation has been structured as follows:In Chapter 1, we introduce the general background of Nevanlinna Theory and some notations which are always used in our studies.In Chapter 2, we investigate meromorphic function sharing one small function with its derivative or its linear differential polynomial. We improve some results of Yu [50] and give partial answers to the questions posed by Yu in the same paper as follows.Theorem 0.1. Let k≥1, f be a nonconstant meromorphic function, and let a be a small meromorphic function of f such that a(z) (?) 0,∞. Suppose thatL(f) = f(k) + ak-1f(k-1)+…+ a0f, (0.0.2)where ak-1,…, a0 are polynomials. If f -a and L(f) - a share the value 0 CM and2δ(0,f)+3(?)(∞,f)>4, then f = L(f).Theorem 0.2. Let k≥1, f be a nonconstant meromorphic function, and let a be a small meromorphic function such that a(z) (?) 0,∞. Suppose that L(f) is given (0.0.2). If f -a and L(f) - a share the value 0 IM and5δ(0, f) + (2k + 6)(?)(∞, f) > 2k + 10, tten f = L(f).We also consider a power of meromorphic function sharing one small function with its derivative and get some results on a conjecture of Brück.Theorem 0.3. Let f be a nonconstant meromorphic (resp. entire) function, n and k be positive integers and a(z) be a small meromorphic function with respect to f such that a(z) (?) 0,∞. If fn-a and (fn)(k) - a share the value 0 CM and n>k + 1 + (?) (resp. n > k + 1), then fn = (fN)(k), and f assumes the formwhere c is a non-zero constant andλk = 1.If a(z)≡1 and / is an entire function in above theorem, we get:Theorem 0.4. Let f be a nonconstant entire function, n and k be positive integers. If fn and (fn)(k)) share 1 CM and n≥k + l, then fN = (fN)(k), and f assumes the form (0.0.3). Theorem 0.4 shows that Brtick Conjecture holds for F = fn, where n≥2 is a positive number and / is a nonconstant entire function. An example given by Gundersen and Yang [16] shows that the assumption n≥2 is sharp.In Chapter 3, two differential polynomials sharing one small function is studied. We improve or extend some previous results given by Hayman, Clunie, Fang and Hua, Yang and Hua, Fang and Qiu, Lin and Yi, and so on. We obtain:Theorem 0.5. Suppose that f is a transcendental meromorphic function with finite number of poles, g is a transcendental entire function, and let n, k be two positive integers with n≥2k+6. If (fn(f- 1))(k) and (gn(g - l))(k) share 1 CM, then f = g.Theorem 0.6. Let f and g be two nonconstant entire functions, and let n,k be two positive integers with n>2k + 4. If(fn)(k) and (gn)(k) share z CM, then either(1) k = 1, f(z) = c1ecz2 , g(z) = c2e-cz2 , where c1, c2 and c are three constants satisfying 4(c1c2)n(nc)2 = -1 or(2) f = tg for a constant t such that tn = 1.In Chapter 4, we do some research on Fermat type functional equation. We will use Nevan-linna Theory to show that there do not exist distinct transcendental meromorphic functions f,g and h that satisfy (0.0.1) for n≥9, which is different from the proof of Hayman's.In Chapter 5, we discuss shared values of meromorphic functions and their shifts. J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo [22] showed that if a finite order function f(z) and f(z + c) share three distinct small periodic functions a1, a2, a3 with period c CM, then f(z) = f(z + c) for all z∈C. We verify that 3CM can be replaced by 2CM+1IM.
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