Meromorphic extendibility and rigidity of interpolation

Let T be the unit circle, f be an α-Hölder continuous function on T, α>1/2, and A be the algebra of continuous function in the closed unit disk that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f+h on T is bigger or equal to −m for any h∈A such that f+h≠0 on T.     

On Montel's theorem and Yang's problem

Let F be a family of meromorphic functions defined in a domain D, and let ψ be a function meromorphic in D. For every function f∈F, if (1)f has only multiple zeros; (2) the poles of f have multiplicity at least 3; (3) at the common poles of f and ψ, the multiplicity of f does not equal the multiplicity of ψ; (4)f(z)≠ψ(z), then F is normal in D. This gives a partial answer to a problem of L. Yang, and generalizes Montel's theorem. Some examples are given to show the sharpness of our result.     

Normal families and shared values of meromorphic functions

Let be a positive integer, let F be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k+1, and let , be two holomorphic functions on D. If, for each f∈F, f=a(z)⇔f(k)=h(z), then F is normal in D.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

Norms of some operators on bounded symmetric domains

Let D be a bounded symmetric domain, H(D) the class of all holomorphic functions on D and u∈H(D). Operator norm of the multiplication operator on the weighted Bergman space , as well as of weighted composition operator from to a weighted-type space are calculated.     

Biholomorphic mappings on bounded starlike circular domains

Let Ω⊂Cn be a bounded starlike circular domain with 0∈Ω. In this paper, we introduce a class of holomorphic mappings Mg on Ω. Let f(z) be a normalized locally biholomorphic mapping on Ω such that and z=0 is the zero of order k+1 of f(z)−z. We obtain a sharp growth theorem and sharp coefficient bounds for f(z). As applications, sharp distortion theorems for a subclass of starlike mappings are obtained. These results unify and generalize many known results.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk

Let D⊂R² be a disk, and let f∈C³. We assume that there is a∈R such that f(a)=0 and f^′(a)>0. In this article, we are concerned with the Neumann problem

     

On deviations and strong asymptotic functions of meromorphic functions of finite lower order

Let f be a transcendental meromorphic function of finite lower order with N(r,f)=S(r,f), and let qν be distinct rational functions, 1?ν?k. For 0<γ<∞ put

     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

On a minimal factorization conjecture

Let be a proper holomorphic map from a connected complex surface S onto the open unit disk D⊂C, with 0∈D as its unique singular value, and having fiber genus g>0. Assume that in case g?2, admits a deformation whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy f∈Mg around ?⁻¹(0) in terms of right-handed Dehn twists induced by the monodromy of has the least number of factors among all possible factorizations of f as a product of right-handed Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585-594]). In this article, the validity of this conjecture is established for g=1.     

Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols

We derive conditions for compactness of Hankel operators () with bounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω⊆Dk.     

Existence of positive solutions for the Hénon equation involving critical Sobolev terms

Let N?3, ^2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation:

     

Normal families and value distribution in connection with composite functions

We prove a value distribution result which has several interesting corollaries. Let k∈N, let α∈C and let f be a transcendental entire function with order less than 1/2. Then for every nonconstant entire function g, we have that (f○g)^(k)−α has infinitely many zeros. This result also holds when k=1, for every transcendental entire function g. We also prove the following result for normal families. Let k∈N, let f be a transcendental entire function with ρ(f)<1/k, and let a₀,…,ak−1,a be analytic functions in a domain Ω. Then the family of analytic functions g such that

     

A system of real quaternion matrix equations with applications

Let H be the real quaternion algebra and H^n×m denote the set of all n×m matrices over H. Let P∈H^n×n and Q∈H^m×m be involutions, i.e., P²=I,Q²=I. A matrix A∈H^n×m is said to be (P,Q)-symmetric if A=PAQ. This paper studies the system of linear real quaternion matrix equations

     

Weighted composition operators from area Nevanlinna spaces into Bloch spaces

Let H(D) denote the class of all analytic functions on the open unit disk D of C. Let φ be an analytic self-map of D and u∈H(D). The weighted composition operator is defined by

     

Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces

Let D be nonempty open convex subset of a real Banach space E. Let be a continuous pseudocontractive mapping satisfying the weakly inward condition and let be fixed. Then for each t∈(0,1) there exists satisfying yt∈tTyt+(1−t)u. If, in addition, E is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if {yt} remains bounded as t→1; in this case, {yt} converges strongly to a fixed point of T as t→¹⁻. Moreover, an explicit iteration process which converges strongly to a fixed point of T is constructed in the case that T is also Lipschitzian.