关于Brück猜想和函数域上的费尔马丢番图方程的研究
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摘要
上世纪20年代，由R．Nevanlinna建立的亚纯函数值分布论(10年后几何形式由L．Ahlfors建立)被认为是关于亚纯函数性质的最重要的成就之一．亚纯函数值分布论在数学的其他领域有着广泛的应用．例如：位势论，多复变，复微分、差分和函数方程，极小曲面等．
亚纯函数唯一性理论伴随着Nevanlinna理论的发展而出现．它主要是研究确定少数函数甚至是一个函数所需要的条件．R．Nevanlinna[39]首先给出了著名的Nevanlinna五值(四值)定理，即两个亚纯函数如果分担扩充复平面上的五个(四个)判别的值，则他们相同(互为线性变换)．这两个定理是唯一性理论的起点，随后又出现了大量的相关结论，如见[44]．除了考虑两个函数之外，也研究亚纯函数与其导数或微分多项式的唯一性，见[44]，第八章，和亚纯函数与其差分的唯一性，见[22]．
众所周知，当n=2，3，4时，费尔马函数方程f(z)~n+g(z)~n+h(z)~n=1存在非常数的亚纯函数解．当n≥9，W．K．Hayman[21]证明费尔马函数方程不存在非常数的亚纯函数解．最近，G．G．Cundersen[14，15]考虑n=5和n=6的情况，用例子验证了费尔马函数方程解的存在性．但是，n=7和n=8的情况还没有解决．
论文的结构如下安排．
第一章，我们介绍Nevanlinna理论和一些经常用的符号．
第二章，我们研究亚纯函数与其导函数或微分多项式分担一个小函数的唯一性，改进了Yu[50]中的一些结果并且回答了Yu提出的部分问题．
定理0．1．设k≥1，f为非常数亚纯函数，a为f的一个小函数满足a(z)≠0，∞．令L(f)=f~((k))+a_(k-1)f~((k-1))+…+a_0f，(0．0．1)其中a_(k-1)，…，a_0为多项式．如果f-a与L(f)-a分担0 CM且2δ(0，f)+3(?)(∞，f)>4，则．f=L(f)．
定理0．2．设k≥1，f为非常数亚纯函数，a为f的一个小函数满足a(z)≠0，∞．令L(f)如(0．0．1)．如果f-a与L(f)-a分担0 IM且5δ(0，f)+(2k+6)(?)(∞，f)>2k+10，则f=L(f)．
我们还考虑了亚纯函数的幂与其导函数分担一个小函数的唯一性，得到了关于Brück猜想的一些结果．
定理0．3．设f为非常数亚纯(整)函数，n和k为正整数，a为f的一个小函数满足a(z)≠0，∞．如果f~n-a与(f~n)~((k))-a分担0 CM且n>k+1+(?)(n>k+1)，则f~n=(f~N)~((k))，且f(z)=ce~(λ/n)z)，(0．0．2)其中c为非零常数，λ~k=1．
在上述定理中，如果a(z)≡1，f为整函数，则
定理0．4．设f为非常数整函数，n和k为正整数．如果f~n与(f~n)~((k))分担1 CM且n≥k+1，则f~n=(f~n)~((k))，且f满足(0．0．2)．
定理0．4表明当F=f~n时，其中n≥2为正整数，f为非常数整函数，Brück猜想成立．由Gundersen and Yang[16]给的一个例子表明上述条件中n≥2是必要的．
第三章研究两个多项式分担一个小函数．我们改进或者推广了由Hayman，Clunie，Fang-Hua，Yang-Hua，Fang-Qiu，Lin-Yi等给出的一些结果．我们得到：
定理0．5．设f为有有限个极点的超越亚纯函数，g超越整函数，n，k为两个正整数满足n≥2k+6．如果(f~n(f-1))~((k))与(g~n(g-1))~((k))分担1 CM，则f=g．
定理0．6．设f和g为两个非常数整函数，n，k为两个正整数满足n>2k+4．如果(f~n)~((k))与(g~n)~((k))分担zCM，则或者
(1)k=1，f(z)=C_1e~(cz~2)，9(z)=c_2e~(-cz~2)，其中c_1，c_2与c为三个常数满足4(c_1c_2)~n(nc)~2=-1或者
(2)f=tg，其中t为常数满足t~n=1．
第四章研究费尔马函数方程．我们将用Nevanlinna理论证明上述费尔马函数方程当n≥9时没有非常数亚纯解．证明方法与Hayman的很不一样．
第五章我们讨论亚纯函数与其平移分担值．J．Heittokangas，R．Korhonen，I．Laineand J．Pdeppo [22]得到如果有穷级的亚纯函数f(z)与f(z+c)分担三个以c为周期的小函数a_1，a_2，a_3 CM，则f(z)≡f(z+c)．我们验证3CM可以被改进为2CM+1IM．
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)) + a_(k-1)f~((k-1))+…+ a_0f, (0.0.2)where a_(k-1),…, a_0 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 f~n-a and (f~n)~((k)) - a share the value 0 CM and n>k + 1 + (?) (resp. n > k + 1), then f~n = (f~N)~((k)), and f assumes the formwhere c is a non-zero constant andλ~k = 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 f~n and (f~n)~((k))) share 1 CM and n≥k + l, then f~N = (f~N)~((k)), and f assumes the form (0.0.3).
Theorem 0.4 shows that Brtick Conjecture holds for F = f~n, 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 (f~n(f- 1))~((k)) and (g~n(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(f~n)~((k)) and (g~n)~((k)) share z CM, then either
(1) k = 1, f(z) = c_1e~(cz~2) , g(z) = c_2e~(-cz~2) , where c_1, c_2 and c are three constants satisfying 4(c_1c_2)~n(nc)~2 = -1 or
(2) f = tg for a constant t such that t~n = 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 a_1, a_2, a_3 with period c CM, then f(z) = f(z + c) for all z∈C. We verify that 3CM can be replaced by 2CM+1IM.
亚纯函数唯一性理论伴随着Nevanlinna理论的发展而出现．它主要是研究确定少数函数甚至是一个函数所需要的条件．R．Nevanlinna[39]首先给出了著名的Nevanlinna五值(四值)定理，即两个亚纯函数如果分担扩充复平面上的五个(四个)判别的值，则他们相同(互为线性变换)．这两个定理是唯一性理论的起点，随后又出现了大量的相关结论，如见[44]．除了考虑两个函数之外，也研究亚纯函数与其导数或微分多项式的唯一性，见[44]，第八章，和亚纯函数与其差分的唯一性，见[22]．
众所周知，当n=2，3，4时，费尔马函数方程f(z)~n+g(z)~n+h(z)~n=1存在非常数的亚纯函数解．当n≥9，W．K．Hayman[21]证明费尔马函数方程不存在非常数的亚纯函数解．最近，G．G．Cundersen[14，15]考虑n=5和n=6的情况，用例子验证了费尔马函数方程解的存在性．但是，n=7和n=8的情况还没有解决．
论文的结构如下安排．
第一章，我们介绍Nevanlinna理论和一些经常用的符号．
第二章，我们研究亚纯函数与其导函数或微分多项式分担一个小函数的唯一性，改进了Yu[50]中的一些结果并且回答了Yu提出的部分问题．
定理0．1．设k≥1，f为非常数亚纯函数，a为f的一个小函数满足a(z)≠0，∞．令L(f)=f~((k))+a_(k-1)f~((k-1))+…+a_0f，(0．0．1)其中a_(k-1)，…，a_0为多项式．如果f-a与L(f)-a分担0 CM且2δ(0，f)+3(?)(∞，f)>4，则．f=L(f)．
定理0．2．设k≥1，f为非常数亚纯函数，a为f的一个小函数满足a(z)≠0，∞．令L(f)如(0．0．1)．如果f-a与L(f)-a分担0 IM且5δ(0，f)+(2k+6)(?)(∞，f)>2k+10，则f=L(f)．
我们还考虑了亚纯函数的幂与其导函数分担一个小函数的唯一性，得到了关于Brück猜想的一些结果．
定理0．3．设f为非常数亚纯(整)函数，n和k为正整数，a为f的一个小函数满足a(z)≠0，∞．如果f~n-a与(f~n)~((k))-a分担0 CM且n>k+1+(?)(n>k+1)，则f~n=(f~N)~((k))，且f(z)=ce~(λ/n)z)，(0．0．2)其中c为非零常数，λ~k=1．
在上述定理中，如果a(z)≡1，f为整函数，则
定理0．4．设f为非常数整函数，n和k为正整数．如果f~n与(f~n)~((k))分担1 CM且n≥k+1，则f~n=(f~n)~((k))，且f满足(0．0．2)．
定理0．4表明当F=f~n时，其中n≥2为正整数，f为非常数整函数，Brück猜想成立．由Gundersen and Yang[16]给的一个例子表明上述条件中n≥2是必要的．
第三章研究两个多项式分担一个小函数．我们改进或者推广了由Hayman，Clunie，Fang-Hua，Yang-Hua，Fang-Qiu，Lin-Yi等给出的一些结果．我们得到：
定理0．5．设f为有有限个极点的超越亚纯函数，g超越整函数，n，k为两个正整数满足n≥2k+6．如果(f~n(f-1))~((k))与(g~n(g-1))~((k))分担1 CM，则f=g．
定理0．6．设f和g为两个非常数整函数，n，k为两个正整数满足n>2k+4．如果(f~n)~((k))与(g~n)~((k))分担zCM，则或者
(1)k=1，f(z)=C_1e~(cz~2)，9(z)=c_2e~(-cz~2)，其中c_1，c_2与c为三个常数满足4(c_1c_2)~n(nc)~2=-1或者
(2)f=tg，其中t为常数满足t~n=1．
第四章研究费尔马函数方程．我们将用Nevanlinna理论证明上述费尔马函数方程当n≥9时没有非常数亚纯解．证明方法与Hayman的很不一样．
第五章我们讨论亚纯函数与其平移分担值．J．Heittokangas，R．Korhonen，I．Laineand J．Pdeppo [22]得到如果有穷级的亚纯函数f(z)与f(z+c)分担三个以c为周期的小函数a_1，a_2，a_3 CM，则f(z)≡f(z+c)．我们验证3CM可以被改进为2CM+1IM．
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)) + a_(k-1)f~((k-1))+…+ a_0f, (0.0.2)where a_(k-1),…, a_0 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 f~n-a and (f~n)~((k)) - a share the value 0 CM and n>k + 1 + (?) (resp. n > k + 1), then f~n = (f~N)~((k)), and f assumes the formwhere c is a non-zero constant andλ~k = 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 f~n and (f~n)~((k))) share 1 CM and n≥k + l, then f~N = (f~N)~((k)), and f assumes the form (0.0.3).
Theorem 0.4 shows that Brtick Conjecture holds for F = f~n, 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 (f~n(f- 1))~((k)) and (g~n(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(f~n)~((k)) and (g~n)~((k)) share z CM, then either
(1) k = 1, f(z) = c_1e~(cz~2) , g(z) = c_2e~(-cz~2) , where c_1, c_2 and c are three constants satisfying 4(c_1c_2)~n(nc)~2 = -1 or
(2) f = tg for a constant t such that t~n = 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 a_1, a_2, a_3 with period c CM, then f(z) = f(z + c) for all z∈C. We verify that 3CM can be replaced by 2CM+1IM.
引文
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[2] H. H. Chen, Yoshida functions and Picard values of integral functions and their derivatives , Bull. Austral. Math. Soc. 54 (1996), 373-381.
[3] Z. X. Chen and K. H. Shon, On conjecture of R. Brück concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), 235-244.
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[5] J. Clunie, On a result of Hayman, J. London Math. Soc. 42 (1967), 389-392.
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[10] G. Frank and G. Weissenborn, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Complex Variables Theory Appl. 7 (1986), 33-43.
[11] G. G. Gundersen, Meromorphic functions that share two finite values with their derivatives , Pacific J. Math. 105 (1983), 299-309.
[12] G. G. Gundersen, When two entire functions and also their first derivatives have the same zeros, Indiana Univ. Math. J. 30 (1981), 293-303.
[13] G. G. Gundersen, Meromorphic functions that share finite values with their derivatives, J. Math. Anal. Appl. 75 (1980), 441-446.
[14] G, G, Gundersen, Meromorphic solutions of f~5 + g~5 + h~5 ≡ 1, Complex Variables Theory Appl. 43 (2001), 293-298.
[15] G, G, Gundersen, Meromorphic solutions of f~6 + g~6 + h~6 = 1, Analysis, Vol. 18 (1998), 285-290.
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[17] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. 31 (2006), 463-478.
[18] R. G. Halburd and R. J. Korhonen, Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477-487.
[19] W. K. Hayman, Meromorphic Functions, Oxford, 1964.
[20] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math. 70 (1959), 9-42.
[21] W, K, Hayman, Waring's Problem fur analytische Funktionen, Bayer. Akad. Wiss. Math. Natur. kl. Sitzungsber, (1985), 1-13.
[22] J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., to appear.
[23] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. L. Zhang, Value sharing results for shifts of meromorphic functions, submitted.
[24] G. Hennekemper, W. Hennekemper, Picard Ausnahmewerte von Ableitungen gewisser meromorpher Funktionen, Complex Variables Theory Appl. 5 (1985), 87-93.
[25] K, Ishizaki, A Note on the Functional Equation f~n + g~n + h~n = 1 and Some Complex Differential Equations, Computat. Methods Funct. Theory, Vol. 2 (1) (2002), 67-85.
[26] G. Jank and N. Terglane, Meromorphic functions sharing three values, Math. Pannon. 2 (2) (1991), 37-46.
[27] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhauser, Basel-Boston, 1985.
[28] I. Lahiri, Uniqueness of a meromorphic function and its derivative, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 1, Art. 20.
[29] I. Lahiri and K. W. Yu, On Generalized Fermat Type Functional Equations, Computat. Methods Funct. Theory 7 (1) (2004), 141-149.
[30] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin-New York, 1993.
[31] I. Laine and C. C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), 148-151.
[32] P. Li and C. C. Yang, When an entire function and its linear differential polynomial share two values, Ill. J. Math. 44 (2000), 349-361.
[33] W. C. Lin, H. X. Yi, Uniqueness theorems for meromorphic functions, Indian J. Pure Appl. Math. 35(2) (2004), 121-132.
[34] W. C. Lin, H. X. Yi, Uniqueness theorems for meromorphic function concering fixedpoints , Complex Var. Theory Appl. 49(11) (2004), 793-806.
[35] L. P. Liu and Y. X. Gu, Uniqueness of meromorphic functions that share one small function with their derivatives, Kodai Math. J. 27 (2004), 272-279.
[36] W. Lü and T. Zhang, On uniqueness of a meromorphic function with its derivative, J. Anal. Appl. 6 (2008), 41-53.
[37] A. Z. Mokhon'ko, The Nevanlinna characteristics of certain meromorphic functions, Teor. Funkci(?) Funkcional. Anal. i Prilozen. 14 (1971), 83-87. (Russian)
[38] E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), 195-206.
[39] R. Nevanlinna, Le thioreme de Picard-Borel et la thiorie des fonctions miromorphes, Gauthier-Villars, Paris, 1929.
[40] M. Ozawa, On the existence of prime periodic entire functions, Kodai Math. Sem. Rep. 29 (1978), 308-321.
[41] L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Lecture Notes in Math. 599 (1977), Springer-Verlag, Berlin, 101-103.
[42] Y. F. Wang, On Mues conjecture and Picard values, Science in China. 36(1) (1993), 28-35.
[43] C. C. Yang, X. H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22(2) (1997), 395-406.
[44] C.C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, The Netherlands, 2003.
[45] L. Z. Yang, Entire functions that share finite values with their derivatives, Bull. Austral. Math. Soc. 41 (1990), 337-342.
[46] P. C. Hu, P. Li and C. C. Yang, Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications, 1. Kluwer Academic Publishers, Dordrecht, 2003.
[47] L. Z. Yang and J. L. Zhang, On some results of Yi and their applications in the functional equations, Acta Math. Sci., revised.
[48] H. X. Yi, Uniqueness theorems for meromorphic functions whose n-th derivatives share the same 1-points, Complex Variables, Vol.34, 421-436.
[49] H, X, Yi, Uniqueness of meromorphic functions and a question of C, C, Yang, Complex Variables Theory Appl. 14 (1990), 169-176.
[50] K. W. Yu, On entire and meromorphic functions that share small functions with their derivatives, JEPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. 21.
[51] J. L. Zhang, Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J., to appear.
[52] J. L. Zhang and L. Z. Yang, Some results related to a conjecture of R. Bru'ck concerning meromorphic functions sharing one small function with their derivatives, Ann. Acad. Sci. Fenn. Math. 32 (2007), 141-149.
[53] J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative , Ann. Acad. Sci. Fenn. Math., to appear.
[54] J. L. Zhang and L. Z. Yang, Some results related to a conjecture of R. Bru'ck, JIPAM. J. Inequal. Pure Appl. Math. 8(2007), no. 1, Art. 18.
[55] J. L. Zhang, Uniqueness theorems for entire functions concerning fixed-points, Comput. Math. Appl., to appear.
[56] J. L. Zhang, A power of an entire function sharing one value with its derivative, submited.
[57] Q. C. Zhang, Meromorphic function that share one small function with its derivative, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 116.