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.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

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.     

Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation

Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e−tf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .     

Stable functions and Vietoris' theorem

An analytic function f(z) in the unit disc D is called stable if s_n(f,·)/f?1/f holds for all for . Here s_n stands for the nth partial sum of the Taylor expansion about the origin of f, and ? denotes the subordination of analytic functions in . We prove that (1−z)^λ, λ∈[−1,1], are stable. The stability of turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials.     

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a=(a₁,…,an) and b=(b₁,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217-222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61-71] (n∈N) for real-valued functions of n variables.     

Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

On controlled extensions of functions

For any numerical function we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions , and satisfy the equality E(f(a),g(a))=h(a), for every a∈A, then we are interested to find the extensions f? and ? of f and g, respectively, such that , for every x∈X. We generalize earlier results concerning E(u,v)=u·v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.     

Dynamics of composite functions meromorphic outside a small set

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E=E(f) and the cluster set of f at any a∈E with respect to is equal to . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f○g and g○f. Denote by F(f) and J(f) the Fatou and Julia sets of f. Let U be a component of F(f○g) and V be a component of F(g○f) which contains g(U). We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.     

Notes on ordered reproducing Hilbert spaces over the complex plane

Let f, g be entire functions. If there exist M₁,M₂>0 such that |f(z)|?M₁|g(z)| whenever |z|>M₂ we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.     

On zero varieties of holomorphic functions in Hardy spaces

In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B_n in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈H^p(B_n), 0<p<∞, with . Here N_g(ζ,1) denotes the integrated zero counting function associated with the slice function g_ζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces H^p(B_n), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.     

A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution to the nonlinear elliptic partial differential equation −∇⋅(a(u(x))∇u(x))=f(x) under periodic boundary conditions, where u(x₀)=u₀ at x₀∈Ω, with Ω=TN, the N-dimensional torus, and N=2,3. The function a is assumed to be smooth, and a(u(x))>0 for , where G⊂R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x₀.     

Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.     

On the Fatou components of two permutable transcendental entire functions

Let f and g be two distinct permutable transcendental entire functions. Suppose further that q(g)=aq(f)+b for some nonconstant polynomial q(z) and constants a(≠0), . In this article, we will investigate the dynamical properties of f and g and show that they have the same Fatou sets with the same components.     

Hensley?s problem for complex and non-Archimedean meromorphic functions

Büchi?s problem asks if there exists a positive integer M such that all x₁,…,xM∈Z satisfying the equations for all 3?r?M must also satisfy for some integer x. Hensley?s problem asks if there exists a positive integer M such that, for any integers ν and a, if ²(ν+r)−a is a square for 1?r?M, then a=0. It is not difficult to see that a positive answer to Hensley?s problem implies a positive answer to Büchi?s problem. One can ask a more general version of the Hensley?s problem by replacing the square by n-th power for any integer n?2 which is called the Hensley?s n-th power problem. In this paper we will solve Hensley?s n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.     

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a₂z²+? satisfying the condition

     

Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays

The paper is concerned with the delay differential equation u^″+λb(t)f(u(t−τ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.     

The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem −Δu=b(x)g(u)+λf(u), u>0, x∈Ω, u_|∂Ω=0, which is independent on λf(u), and we also show the existence and uniqueness of solutions to the problem, where Ω is a bounded domain with smooth boundary in RN, λ>0, g∈C¹((0,∞),(0,∞)) and there exists γ>1 such that , ∀ξ>0, , the function is decreasing on (0,∞) for some s₀>0, and b is nonnegative nontrivial on Ω, which may be vanishing on the boundary.