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1.
A criterion of normality based on a single holomorphic function 总被引:1,自引:0,他引:1
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f
(k)(z)| ≤ c, where c is a constant. Then F is normal on D. 相似文献
2.
Let G ⊂ C be a domain which is bounded by a finite number of pairwise disjoint Jordan curves. We prove the existence of a function
which is holomorphic exactly on G and has universal translates with respect to a prescribed set E ⊂∂G and which in addition is continuous on G
-\E.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
Yoshinori Hamahata 《manuscripta mathematica》1993,79(1):307-327
We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym2(ϕ), and ∧2(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F
q
(T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ. 相似文献
4.
Michael Koren 《Israel Journal of Mathematics》1973,15(4):396-403
It is shown that the realizability of the sequences ϕ=(a
1,…,
a
), ψ=(b
1,…,b
n
) and ϕ+ψ is a sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor ifb
i
≦1 fori=1,…,n. The condition is not sufficient in general. A necessary and sufficient condition for the realizability of ϕ+ψ by a graph
with a ϕ-factor is given for the case that ϕ is realizable by a star and isolated vertices. 相似文献
5.
Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C
ϕ
, C
ψ
the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition
operators C
ϕ
−C
ψ
from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized. 相似文献
6.
Ngaiming Mok 《中国科学A辑(英文版)》2008,51(4):660-684
In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures. 相似文献
7.
Piotr Kot 《Czechoslovak Mathematical Journal》2009,59(2):371-379
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.
相似文献
8.
Josip Globevnik 《Israel Journal of Mathematics》2000,115(1):195-203
Given a bounded convex domainD in ℂN with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely. 相似文献
9.
Ricardo Abreu Blaya Juan Bory Reyes Tania Moreno García 《Bulletin of the Brazilian Mathematical Society》2007,38(4):635-647
We consider the problem of identifying boundary values of holomorphic functions on bounded domains in ℂ2. We use the quaternionic analysis techniques to extending the CR structure to a pure function theoretical nature. The advantage of our procedure lies in the fact that it also runs for domains
with fractal boundary. 相似文献
10.
The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces
Erhan alkan 《Mathematische Nachrichten》2006,279(7):705-715
We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
For ϕ a δ-subharmonic function, sharp results are obtained that connectA(r, ϕ), B(r, ϕ) andT(r, ϕ), whereA(r, ϕ)=inf|z|=r
ϕ(z),B(r, ϕ)=sup|z|=r
ϕ(z), andT(r, ϕ) is the Nevanlinna characteristics. 相似文献
12.
For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H
0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H
1 : f∈Λɛ where Λɛ is a set of functions on R
1 of the form Λɛ = {f : f∈?, ϕ(f) ≥ Cψɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C
* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C
* and is not possible for C < C
*. We propose asymptotically minimax test statistics.
Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000 相似文献
13.
Piotr Kot 《Czechoslovak Mathematical Journal》2007,57(1):29-47
We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball
For a function u which is lower semi-continuous on
we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈
such that
. 相似文献
14.
Ngaiming MOK 《数学年刊B辑(英文版)》2008,29(2):135-146
The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f:Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′:=rank(Ω′)≤rank(Ω):=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f:Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω→Ω′) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case. 相似文献
15.
Walter Bergweiler 《Journal d'Analyse Mathématique》1994,63(1):121-129
Let (zj) be a sequence of complex numbers satisfying |zj|→ ∞ asj → ∞ and denote by n(r) the number of zj satisfying |zj|≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫∞ (ϕ(t)t logt)−1
dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here.
These results answer a question by A. A. Gol’dberg. 相似文献
16.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f
(k)(z)| < |h(z)|; (b) f
(k)(z) ≠ h(z). Then F is normal on D. 相似文献
17.
Joseph Weier 《Annali dell'Universita di Ferrara》1958,8(1):29-37
Riassunto SeM edN sono varietà poliedriche chiuse connesse ed orientate di dimensioni rispettivem edn, conm≥n>2, edf∶M→N è una trasformazione continua, allora per ognir, minore din e non inferiore a 2, si definisce un omomorfismo indotto ϕrπ:r
(N)→H
m-n+r
(M) dal quale si ricavano certi invarianti topologici.
Résumé Soientm≥n>r≥2 des entiers etM, N des variétés polyédrales closes connexes orientées satisfaisant dimM=m et dimN=n, de plusH i(M) le groupe de Betti à i dimensions deM,M,π i (N) le groupe de Hurewicz ài dimensions deN, etf∶M→N une application continue. Alorsf définit, pour,r=2, 3, …n−1, un homomorphisme réciproque ϕrπ:r (N)→H m-n+r (M) comme il suit. Etant donné un élément α du groupe πr (N) et uner-sphère continue orientéeS de α, on peut supposer quef −1(S) soit un polyèdre finiA àm−n+r dimensions. Parf est induit dansA un (m−n+r)-cyclez à coefficients entiers, et la classe d'homologie dez est justement l'image ϕr(α) de α par ϕr. Pourr=1, on obtient un homomorphisme réciproque ϕrπ:r (N)→H m-n+r (M) du groupe fondamentalF(N) deN dans le groupe d'homologie àm−n+1 dimensions deM. A l'aide des homomorphismes ϕ,,ϕ2,ϕ,3...,ϕn-i, on parvient à certaines expressions caractéristiques dépendantes seulement de la classe d'homotopie def, en particulier on obtient des constantes pour les images des bases de Betti deM, pour Fimage du groupe de torsion deM, et pour l'image réciproque du groupe fondamental deN.相似文献
18.
B. I. Peleshenko 《Ukrainian Mathematical Journal》2005,57(11):1741-1762
We prove theorems on interpolation of quasilinear operators of weak type (ϕ0, ψ0, ϕ0, ψ1) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and
convex functions ϕ0(t), ψ0(t), ϕ1(t), and ψ1(t).
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005. 相似文献
19.
Assume thatf is an integer transcendental solution of the differential equationP
n
(z, f, f′)=P
n−1(z, f, f′, ... f
(p)), whereP
n
andP
n−1 are polynomials in all variables, the degree ofP
n
with respect tof andf′ is equal ton, and the degree ofP
n−1 with respect tof, f′, ... f
(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E
*, whereE
* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z
1/2 (β+o(1)), β∈C, holds forz=re
iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re
iν)‖=o(r
1/2),r→+∞,r>0,
, where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77,
January, 1999. 相似文献
20.
D. V. Prokhorov 《Mathematical Notes》1997,61(5):609-613
We solve the maximal value problem for the functional
in the class of functionsf(z)=z+a
2z2+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk
1,…,km contains an even number.
Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997.
Translated by S. S. Anisov 相似文献