Spherical derivatives and normal families

We show that a family of functions meromorphic in a plane domain D whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each f meromorphic in the unit disk D, inf _z∈D f ^#(z) ≤ 1/2, where f ^# denotes the spherical derivative of f.     

Nonvanishing theorems for L-functions of modular forms and their derivatives

Inventiones mathematicae -     

Periodic points and normal families

Let be the family of all functions which are holomorphic in some domain and do not have periodic points of some period greater than one there. It is shown that is quasinormal, and the sequences in which do not have convergent subsequences are characterized. The method also yields a new proof of the result that transcendental entire functions have infinitely many periodic points of all periods greater than one.

     

Differential polynomials and normal families

This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu's criterion for the normality of a family of meromorphic functions.     

非零单叶函数的极值问题

1980年,P.Duren和G.Schober引入了非零单叶函数族S_0,f∈S_0是指f(z)在单位园盘内单叶解析,恒不等于零并且用条件f(0)=1正规化。在后来的一系列文章中,他们研究了S_0的闭凸包HS_0中的线性极值问题。设ψ是S_0上的实值连续凸泛函,本文研究了凸极值问题maxψ(f)f∈S_0,得到极值函数值域及其不取弧的一些性质,这些结果包含了P.Duren和G.Schober的结果。     

Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z ₀) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) ? α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β ₀ and α(z) ? α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.     

Surface Bundles with Nonvanishing Signature

Using the theory of Lefschetz fibrations and recent advances in mapping class group theory, surface bundles over surfaces with nonzero signature and small base genus are constructed. In particular, a genus-5 fibration over the surface of genus 26 with nonzero signature is given –- improving former results on the possible base genera for surface bundles over surfaces with nonzero signature.     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditionsall zeros of f(z)-z are multipleandf~k having at most k distinct fixed points in Dare the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditionsl 3for k 3andf~k having at most one fixed point in Dare the best possible.     

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n_M such that for all finitely generated Λ-modules N, if Ext_Λⁱ(M,N)=0 for all i?0, then Ext_Λⁱ(M,N)=0 for all i?n_M. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.     

On normal families of meromorphic functions

In this paper, we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions, which improve and generalize the related results of Gu, Bergweiler and Lin.     

On normal families of meromorphic functions

T. Qian In this paper, we prove two theorems on normal families of meromorphic functions, which improve a few of results from several authors. Copyright © 2015 John Wiley & Sons, Ltd.     

Fixed points and normal families of quasiregular mappings

LetF be a family of mappingsK-quasiregular in some domainG. We show that if for eachf∈F, there existsk>1 such that thek-th iteratef ^k off has no fixed point, thenF is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove thatF is quasinormal, ifF contains only quasiregular mappings that do not have periodic points of some period greater than one inG. This implies that a quasiregular mappingf:ℝ ⁿ with an essential singularity in ∞ has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.     

Share fix-points and normal families of holomorphic functions

Let be a family of holomorphic functions in a domain D, let k 2 be a positive integer, and let K be a positive number. In this paper, we prove that: if, for each f and f share fix-points CM, and |f^(k) (z)| K whenever f(z) = z in D, then is normal in D. Some examples are given to show the sharpness of our result.Received: 17 June 2004     

Bloch radius, normal families and quasiregular mappings

Bloch's Theorem is extended to -quasiregular maps , where is the standard -dimensional sphere. An example shows that Bloch's constant actually depends on for .

     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

Simultaneous Nonvanishing of Twists of Automorphic L-Functions

Given three distinct primitive complex characters ₁,₂,₃ satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·₁,1/2) L(f·₂,1/2) L(f·₃,1/2) does not vanish for a positive proportion of weight 2 primitive forms for ₀(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J ₀(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.P.M. is partially supported by NSF Grant DMS-97-2992 and by the Ellentuck fund (by grants to the Institute for Advanced Study) and by the Institut Universitaire de France.