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1.
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 f∈F, 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
2.
Yuntong Li 《Journal of Mathematical Analysis and Applications》2011,381(1):344-351
Let F be a family of meromorphic functions defined in a domain D such that for each f∈F, all zeros of f(z) are of multiplicity at least 3, and all zeros of f′(z) are of multiplicity at least 2 in D. If for each f∈F, f′(z)−1 has at most 1 zero in D, ignoring multiplicity, then F is normal in D. 相似文献
3.
Pei-Chu Hu 《Journal of Mathematical Analysis and Applications》2009,357(2):323-731
Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D. 相似文献
4.
We prove a uniqueness theorem for non-constant meromorphic functions f, g which share three values 0, 1, ∞ and f−a, g−b share the value 0 for a,b∉{0,1,∞}. Our theorem improves a result of G. Brosch. 相似文献
5.
Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k + 1, and f + a(f^(k))^n≠b in D, then F is normal in D. 相似文献
6.
Let k be a positive integer and F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k+1. If for each pair (f, g) in F, ff(k) and gg(k) share a non-zero complex number a ignoring multiplicity, then F is normal in D. 相似文献
7.
Qian Lu 《Journal of Mathematical Analysis and Applications》2008,340(1):394-400
We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε0 such that |an(a(z)−1)−1|?ε0 for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f′(z)|?|f(z)| whenever f(z)f″(z)−a(z)(f′2(z))=0 in Δ. Then is normal in Δ. 相似文献
8.
Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let F{\mathcal{F}} be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in F, P(f)f(k){\mathcal{F}, P(f)f^{(k)}} and P(g)g
(k) share b in D, then F{\mathcal{F}} is normal in D. 相似文献
9.
In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D■C,a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f ∈ F , there exists g ∈ G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D. 相似文献
10.
In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM∗, and share b(z)IM∗ with g(z)=a1(z)f(z)+a2(z)f′(z). And a1(z), a2(z) and b(z) (a2(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution. 相似文献
11.
Alain Escassut Jacqueline Ojeda 《Journal of Mathematical Analysis and Applications》2009,351(1):350-359
Recently, in the complex context, several results were obtained concerning functional equations of the form P(f)=Q(g) where P and Q are polynomials of only two or three terms whose coefficients are small functions: in certain cases the equation does not admit any pair of admissible solutions and in other cases it only admits pairs of solutions that are of a very particular type. Here we consider similar questions when the ground field is a p-adic complete algebraically closed field of characteristic 0 and we derive results that are often analogous. For instance, if fn+a1fn−m+b1=c(g−n+a2gn−m+b2), with ai, bj small functions with regard to f, g and a2b2 non-identically 0, then and f=hg with . However, contrary to the complex context, here results apply not only to meromorphic functions defined in the whole field but also to unbounded meromorphic functions defined inside an open disc. The main tool is the p-adic Nevanlinna theory. 相似文献
12.
Let K be a complete ultrametric algebraically closed field of characteristic π. Let P,Q be in K[x] with P′Q′ not identically 0. Consider two different functions f,g analytic or meromorphic inside a disk |x−a|<r (resp. in all K), satisfying P(f)=Q(g). By applying the Nevanlinna's values distribution Theory in characteristic π, we give sufficient conditions on the zeros of P′,Q′ to assure that both f,g are “bounded” in the disk (resp. are constant). If π≠2 and deg(P)=4, we examine the particular case when Q=λP (λ∈K) and we derive several sets of conditions characterizing the existence of two distinct functions f,g meromorphic in K such that P(f)=λP(g). 相似文献
13.
Normality and shared values 总被引:19,自引:0,他引:19
LetF be a family of meromorphic functions on the unit disc Δ and leta andb be distinct values. If for everyf∈F,f andf′ sharea andb on Δ, thenF is normal on Δ.
The first author was supported by NNSF of China approved no. 19771038 and by the Research Institute for Mathematical Sciences,
Bar-Ilan University. 相似文献
14.
In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results. 相似文献
15.
Nguyen Thi Thu Hang 《Journal of Mathematical Analysis and Applications》2011,376(2):656-674
A well-known result of Nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane C and for four distinct values aj∈C∪{∞}, if νf−aj=νg−aj for all 1?j?4, then g is a Möbius transformation of f. In 1983, Gundersen generalized the result of Nevanlinna to the case where the above condition is replaced by: min{νf−aj,1}=min{νg−aj,1} for j=1,2 and νf−aj=νg−aj for j=3,4. In this paper, we prove that the theorem of Gundersen remains valid to the case where min{νf−aj,1}=min{νg−aj,1} for j=1,2, and min{νf−aj,2}=min{νg−aj,2} for j=3,4. Furthermore, we work on the case where {aj} are small functions. 相似文献
16.
Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D. 相似文献
17.
In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k+12. If fn + af(k) and gn + ag(k) share b CM and the b-points of fn + af(k) are not the zeros of f and g, then f and g are either equal or closely related. 相似文献
18.
P. Niu Y. Xu 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2016,51(3):160-165
The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f(k) and g(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities mf for f and mh for h satisfy mf ≥ mh + k + 1 for k > 1 and mf ≥ 2mh + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy nf ≥ nh + 1, then the family F is normal on D. 相似文献
19.
Normal families of meromorphic functions with multiple values 总被引:1,自引:0,他引:1
Jiying Xia 《Journal of Mathematical Analysis and Applications》2009,354(1):387-393
Let F be a family of meromorphic functions defined in a domain D, let ψ(?0) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f∈F, f≠0, f(k)≠0, and all zeros of f(k)−ψ(z) have multiplicities at least (k+2)/k. If, for k=1, ψ has only zeros with multiplicities at most 2, and for k?2, ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al. 相似文献
20.
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) = c1ecz2, g(z) = c2e− cz2, where c1, c2, 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