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1.
This paper is devoted to proving some uniqueness type results for an entire function f(z) that shares a common set with its shift f(z+c) or its difference operator Δcf. We also give some applications to solutions of non-linear difference equations related to a conjecture proposed by C.C. Yang.  相似文献   

2.
We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value aC infinitely often. This research also shows that there exist two sets S1 with 9 (resp. 5) elements and S2 with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.  相似文献   

3.
Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + 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 ∈ 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.  相似文献   

4.
Growth of solutions of second order linear differential equations   总被引:1,自引:0,他引:1  
This paper is devoted to studying the growth of solutions of equations of type f+h(z)eazf+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.  相似文献   

5.
In this paper, we prove the following result: Let f(z) and g(z) be two nonconstant meromorphic(entire) functions, n ≥ 11(n ≥ 6) a positive integer. If fn(z)f′(z) and gn(z)g′(z) have the same fixed-points, then either f(z) = c1ecz2g(z) = c2e− cz2, where c1c2, and c are three constants satisfying 4(c1c2)n + 1c2 = −1, or f(z) ≡ tg(z) for a constant t such that tn + 1 = 1.  相似文献   

6.
Estimates for the zeros of differences of meromorphic functions   总被引:6,自引:0,他引:6  
Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.  相似文献   

7.
Let f(z) be a holomorphic function in a hyperbolic domain Ω. For 2?n?8, the sharp estimate of |f(n)(z)/f(z)| associated with the Poincaré density λΩ(z) and the radius of convexity ρΩc(z) at zΩ is established for f(z) univalent or convex in each Δc(z) and zΩ. The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev-Wirths conjecture is also discussed.  相似文献   

8.
Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a(z) or b(z)?b(z). In this paper, we prove that: if, for each fF, f(z)−a(z) and f(z)−b(z) have no common zeros, f(z)=a(z) whenever f(z)=a(z), and f(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.  相似文献   

9.
In this paper, we study the differential equations of the following form w2+R(z)2(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w2+P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w2+(zz0)P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2+A(zz0)2(w)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab2=1, a2=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B2=C and q(z)=±P(z).  相似文献   

10.
We study the differential equations w 2+R(z)(w (k))2 = Q(z), where R(z),Q(z) are nonzero rational functions. We prove
  1. if the differential equation w 2+R(z)(w′)2 = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then QC (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
  2. if the differential equation w 2 + R(z)(w (k))2 = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, QC (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .
  相似文献   

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