Meromorphic Functions Sharing Values And Value Distribution Of Some Differential And Difference Equations

In1920s, R. Nevanlinna introduced the characteristic functions of meromor-phic functions, and proved two fundamental theorems, thereby establishing the value distribution theory or so called Nevanlinna theory. This theory is consid-cred to be one of the most important achievements in the20th century. It plays a fundamental role for modern researches on meromorphic function theory and has a profound influence on the development of many other mathematical branches. In spite of the relative completeness of the value distribution theory, the study of even some classical problems has not been brought to an end. On the con-trary, it becomes more and more extensive. Moreover, Nevanlinna theory and its extensive are having numerous applications in various fields of mathematics, for example, potential theory, normal family, several complex variables theory, complex differential and difference equations, and so on.It is a common consensus that any polynomial is determined by its zero points (the set on which the polynomial take zeros) except for a non-constant factor, but it is not true for transcendental entire function and meromorphie function. Therefore, how to determine a meromorphic function becomes inter-esting and complex. The uniqueness theory of meromorphic functions mainly studies conditions under which there exists essentially only one function satisfy-ing these conditions and relations between functions satisfying given conditions. R. Nevanlinna himself proved the classic results of the uniqueness theory, like five-value and four-value theorems. Later on, many mathematicians have studied deeply into this field. During the last two decades, this theory has developed to various directions, such as uniqueness studies of a meromorphic function and its derivative, see [1].As a recent analogue, Hcittokangas [2] ct al. started to consider the unique-ness of a moromorphic function sharing values with its shift. The background of these considerations lies in the recent difference counterparts of Nevanlinna the-ory. The key result here is the difference analogue of the lemma on the logarithmic derivative obtained by Halburd, Korhoncn [3,4], Chiang, Feng [5] respectively.In the present thesis, we investigated the uniqueness of mcromophic functions sharing two values and the value distribution of some complex differential and difference equations. It consists of four parts and the matters are explained as below.In Chapter1, we introduced some classic results of Nevanlinna Theory, the difference analogue of the lemma on the logarithmic derivative and the Wiman-Valiron theorem.In Chapter2, we investigated the uniqueness problem of meromophic func-tions sharing two values, obtaining some uniqueness theorems, which improve the results given by Fang and Hua [6], Yang and Hua [7], Fang and Qiu [8]. In fact, we obtained the following results.Theorem0.1. Let f and g be two transcendental meromorphic functions, whose zeros are of multiplicities at least k, where k is a positive integer. Let n> max{2k-1, k+A/k+4} be a positive integer. If f(k) and gn(k) share z CM, f and g share∞IM, one of the following two conclusions holds:(i)fnf(k)＝gng(k).(ii) f(z)=c1ecz2g(z)=c2e-cz2, where C1,c2and c are constants such that4(c1c2)n+1c2=-1.Theorem0.2. Let f and g be two non-constant meromorphic functions, whose zeros are of multiplicities at least k, where k is a positive integer. Let n> max(2k-1, k+4/k+4} be a positive integer. If fnf(k) and gng(k) share1CM, f and g share oo IM, one of the following two conclusions holds:(i)fnf(k)=gng(k). (ii)f(z)=c3edz,g(z)=c4e-dz,where c3,c4and d are constants such that (-1)k(c3c4)n+1d2k=1.In Chaptcr3,we investigated the properties of cntirc solutions of difference equations and difference polynomials. Meanwhile,some examples were given show that the conditions of the results are necessary.We obtained the following results.Theorem0.3. Letp(z), r(z) and s(z) be polynomials, not vanishing. Ifn> m+1(or m> n+1) are two positive integers, then the non-linear difference equation f(z)n+p(z)f(z+c)m=r(z)es(z), has no transcendental entire solutions of finite order.Theorem0.4. Consider the difference equation f(z)n+p(z)f(z+c)m=q(z), where p and q are non-zero entire, functions with finite order, m and n are positive integers, c is a non-zero complex constant. If entire function f satisfies that λ(f)<σ(f)=∞and σ2(f)<∞; then f is not a solution of the equation.Theorem0.5. Suppose that the equation f(z)n+p(z)f(z+c)m=q(z) has an entire function f with finite order, where q are non-zero entire functions with finite order and p is a small function off, m7^n are positive integers, c is a non-zero complex constant. Then σ(f)=σ(q).Corresponding to fnfi, we investigated the value distribution of f(z)n (f(z)-1)f(z+c), solving the problem which was not dealt with by Zhang [9] and Qi [10] that n=1. We obtained the following results. heorem0.6. Let f be a transcendental entire function of finite order with Borel exceptional value a, and let c be a non-zero complex constant. Then λ(f(f-1)f(z+c)-b)=σ(f), forb≠a3-a.Theorem0.7. Let f be a transcendental entire function of finite order and let c be a non-zero complex constant. If f(z) or f(z)-1has infinitely many multi-ordei zeros, then f(z)(f(z)-1)f(z+c) assumes every value a￡C infinitely often.In Chapter4,we investigated the value distribution of some differential and difference equations related to Briick conjecture, improving the results given by Chen, Shon [11], Wang [12] and Liu, Chen [13]. We obtained the following main theorems.Theorem0.8. Let f be a non-constant entire function,σ2ν)(<∞)is not a positive integer. Set where aj(z)(2<j≤k) are entire functions of order less than1and ak(z)(?)0. If f and Li(f) share z IM, and then L1(f)-z=h(z)(f-z), where h is a meromorphic function of order no greater than s.Theorem0.9. Let f be a non-constant entire function, σ(f)<1/2. Let a be a non-zero small function of f. Set wheer aj(z)(j=0,1,...,k)are polymomial and ak(z)(?)0.If then where B(z)is a non-zero polynonial.