The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A sim...The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A simple andelementary way to study such subjects is exhibited in this paper; that is,once an estimation of B(r,*) in terms of a few C(r,**) in the Nevanlinna theory on angular domains is established, we can produce one result that the order of a mermorphic function with radially distributed values related to C(r,**) can be estimated under the assumption of existence of suitable deficient value. The results obtained in this paper lead us to a new singular direction in terms of Nevanlinna characterstic instead of the order of meromorphic functions.展开更多
Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is k...Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.展开更多
For a meromorphic function f, let N（l＋1（r, 3） denote the counting function of zeros of f of order 1 at least. Let f be a nonconstant meromorphic function, such that N（r,f） =S（r,f）. Denote F = fn. Suppose that ...For a meromorphic function f, let N（l＋1（r, 3） denote the counting function of zeros of f of order 1 at least. Let f be a nonconstant meromorphic function, such that N（r,f） =S（r,f）. Denote F = fn. Suppose that F andF＇ share 1 CM. If（l） n 〉 3, or （2） n = 2 and N（r, 3） = O（N（3（r, 1/f））, then, F = F＇, and f assumes the form l（z） = ce 1/nz where c is a nonzero constant. This main result of this article gives a positive answer to a question raised by Zhang and Yang [1] for the meromorphic functions case in some sense. And a relative result is proved.展开更多
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.19971049)BRF of Tsinghua University.
文摘The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A simple andelementary way to study such subjects is exhibited in this paper; that is,once an estimation of B(r,*) in terms of a few C(r,**) in the Nevanlinna theory on angular domains is established, we can produce one result that the order of a mermorphic function with radially distributed values related to C(r,**) can be estimated under the assumption of existence of suitable deficient value. The results obtained in this paper lead us to a new singular direction in terms of Nevanlinna characterstic instead of the order of meromorphic functions.
基金supported by the China Scholarship Council (CSC)supported in part by the Academy of Finland #121281
文摘Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.
基金supported by NNSF of China(11171013)Fundamental Research Funds for the Central Universities NO.300414supported by the Innovation Foundation of BUAA for Ph.D.Candidates
文摘For a meromorphic function f, let N（l＋1（r, 3） denote the counting function of zeros of f of order 1 at least. Let f be a nonconstant meromorphic function, such that N（r,f） =S（r,f）. Denote F = fn. Suppose that F andF＇ share 1 CM. If（l） n 〉 3, or （2） n = 2 and N（r, 3） = O（N（3（r, 1/f））, then, F = F＇, and f assumes the form l（z） = ce 1/nz where c is a nonzero constant. This main result of this article gives a positive answer to a question raised by Zhang and Yang [1] for the meromorphic functions case in some sense. And a relative result is proved.
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.
