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1.
Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann. 相似文献
2.
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
3.
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
4.
Jilong Zhang 《Journal of Mathematical Analysis and Applications》2010,367(2):401-490
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 a∈C 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
5.
Kamal Boussaf 《Bulletin des Sciences Mathématiques》2010,134(1):44
Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)n∈N in a disk d(0,R−) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions. 相似文献
6.
Zong-Xuan Chen 《Journal of Mathematical Analysis and Applications》2008,344(1):373-383
Let f be a transcendental meromorphic function and g(z)=f(z+1)−f(z). A number of results are proved concerning the existences of zeros and fixed points of g(z) or g(z)/f(z) which expand results of Bergweiler and Langley [W. Bergweiler, J.K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007) 133-147]. 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.
9.
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
10.
Risto Korhonen 《Journal of Mathematical Analysis and Applications》2009,357(1):244-253
A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if n∈N and three distinct values of a meromorphic function f of hyper-order less than 1/n2 have forward invariant pre-images with respect to a fixed branch of the algebraic function τ(z)=z+αn−1z1−1/n+?+α1z1/n+α0 with constant coefficients, then f○τ≡f. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images. 相似文献
11.
Jia-Feng Tang 《Journal of Mathematical Analysis and Applications》2007,334(1):517-527
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+(z−z0)P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2+A(z−z0)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
12.
Xiaoling Wang Chung-Chun Yang 《Journal of Mathematical Analysis and Applications》2006,324(1):373-380
We investigate the factorization of entire solutions of the following algebraic differential equations:
bn(z)finjn(f′)+bn−1(z)fin−1jn−1(f′)+?+b0(z)fi0j0(f′)=b(z), 相似文献
13.
An even-order three-point boundary value problem on time scales 总被引:1,自引:0,他引:1
Douglas R Anderson Richard I Avery 《Journal of Mathematical Analysis and Applications》2004,291(2):514-525
We study the even-order dynamic equation (−1)nx(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x(Δ∇)i(a)=0 and x(Δ∇)i(c)=βx(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale. 相似文献
14.
Khalida Inayat Noor 《Journal of Mathematical Analysis and Applications》2003,281(1):244-252
Integral operator, introduced by Noor, is defined by using convolution. Let fn(z)=z/(1−z)n+1, n∈N0, and let f be analytic in the unit disc E. Then Inf=f(−1)n★f, where fn★f(−1)n=z/(1−z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense. 相似文献
15.
On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function
In this paper we study the distribution of zeros of each entire function of the sequence , which approaches the Riemann zeta function for Rez<−1, and is closely related to the solutions of the functional equations f(z)+f(2z)+?+f(nz)=0. We determine the density of the zeros of Gn(z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip. 相似文献
16.
Qingcai Zhang 《Journal of Mathematical Analysis and Applications》2008,338(1):545-551
In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f′−afn and g′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3. 相似文献
17.
Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz2?1(z) and f(z)=e−βz2f1(z), where ?1 and f1 have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation. 相似文献
18.
On Homogeneous Differential Polynomials
of Meromorphic Functions 总被引:2,自引:0,他引:2
In this paper, we study one conjecture proposed by W. Bergweiler and show that any
transcendental meromorphic functions f(z) have the form exp(αz+β) if f(z)f″(z)–a(f′ (z))2≠0,
where
. Moreover, an analogous normality criterion is obtained.
Supported by National Natural Science Foundation and Science Technology Promotion Foundation of Fujian
Province (2003) 相似文献
19.
K. A. Narayanan 《Proceedings Mathematical Sciences》1974,80(2):75-84
Letf(z) be meromorphic function of finite nonzero orderρ. Assuming certain growth estimates onf by comparing it withr ρ L(r) whereL(r) is a slowly changing function we have obtained the bounds for the zeros off(z) ?g (z) whereg (z) is a meromorphic function satisfyingT (r, g)=o {T(r, f)} asr → ∞. These bounds are satisfied but for some exceptional functions. Examples are given to show that such exceptional functions exist. 相似文献
20.
Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑n?0anz-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived. 相似文献