Nonconstant Analytic Function Research Articles

On zero sets of harmonic and real analytic functions

In this paper we study some questions related to the zero sets of harmonic and real analytic functions in $${\mathbb {R}}^N$$ . We introduce the notion of analytic uniqueness sequences and, as an application, we show that the zero set of a non-constant real analytic function on a domain always has empty fine interior. We also prove that, for a certain category of sets $$E\subset {\mathbb {R}}^N$$ (containing the finely open sets), each function f defined on E is the restriction of a real analytic (respectively harmonic) function on an open neighbourhood of E if and only if f is “analytic (respectively harmonic) at each point” of E.

Computation of the Łojasiewicz exponent of nonnegative and nondegenerate analytic functions

Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for ‖x‖ ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by [Formula: see text], is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent [Formula: see text] of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate.

Generation and commutation properties of the Volterra operator

Let V be the classical Volterra operator on L 2(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert–Schmidt operators and as well in the space \({\mathcal{B}(L_2(0,1))}\) equipped with the weak operator topology. Moreover, the algebra generated by V 2 and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.

A CHARACTERIZATION OF NONLINEAR OPEN MAPPINGS BETWEEN PSEUDOMETRIZABLE TOPOLOGICAL VECTOR SPACES

We prove a new characterization of arbitrary (including nonlinear) open maps between pseudometrizable topological vector spaces. We use it to give a new proof that nonconstant analytic functions are open.

On limit functions and their natural boundaries in infinite compositions of entire functions

In this article, we consider infinite sequences {Φ n } of entire functions in the complex plane C defined as compositions of the form where each f n , n = 1, 2, … , is an entire function, and the limit functions Φ of such sequences. Under reasonable conditions on the sequence {f n }, and for the cases where Φ exists and is a nonconstant analytic function, one finds that the boundary of the domain where {Φ n } converges to Φ is in fact the natural boundary of Φ, and that this boundary satisfies certain “expansion” properties when considered under the composition of the f n 's. We also consider the case of constant limit functions Φ. In the final section we discuss the connection between the coefficients of a power series representation of a nonconstant limit Φ and the sequence {a n } of a one parameter family of entire functions f n (z) = f (a n,z ), whose composition as in (1) converges in some domain to Φ.

A proper subclass of Maclane′s class 𝒜

The MacLane′s class 𝒜 of analytic functions is the class of nonconstant analytic functions in the unit disk that have asymptotic values at a dense subset of the unit circle. In this paper, we define a subclass ℛ of 𝒜 consisting of those functions that have asymptotic values at a dense subset of the unit circle reached along rectifiable asymptotic paths. We also show that the class ℛ is a proper subclass of 𝒜 by constructing a function f ∈ 𝒜 that admits no asymptotic paths of finite length.

On Composition Operators on N+(?)

Let N(?) denote the class of analytic functions fin a domain ?, contained in the complex numbers C, such that log(1+| f |) has a harmonic majorant. The subclass N+(?) of N(?) consists of all f such that log(1+| f |) has a quasi-bounded harmonic majorant. Let ? be a non-constant analytic function from ? into itself Define the composition operator C?, on N(?) by C?f=fo?, V f € N(?). Then C?, maps N+(?) into itself. Here we characterize the invertibility of C? when ? is finitely connected with boundary ? consisting of disjoint analytic simple closed curves and we give a necessary condition for the density of the range of C?, in N+(?). Moreover, we consider linear isometries on N+(?) and their relation to C?.

Juiia Sets and Non-Constant Limits in the Composition of Entire Functions

With any given sequence of entire functions, we associate the sequence of entire functions defined by The set of normality Simply N if it is clear to which sequence we are referring), will mean the set of all points in a where only is a normal a neighbourhood U of such that every subsequencc of contains a subsequence which converges locally uniformlv in U either lu a finite limit function or to the constant limit ∞. The complement of A in C will be denoted by or hy (J) and called the Tulia set In this paper we consider sequences which converge locally uniformly in. the plane to the identity function, such that the corresponding sequence converges in some domain to a non-constant analytic function, and study the maximal domains where this process of composition converges to these functions. Every such domain is in fact a component of We also study the behaviour of in other domains in C, where it does not converge to a non-constant limit.

John constants for polynomials

Let g be some set of non-constant analytic functions in the unit disk D, and for f e dg define The quantity where g u is the set of univalent functions in g, is called the John constant of g. We discuss where T k consists of the trinomials In particular, we find We also establish a conjecture of Rahman and Szynal for univalent trinomials in T k

Angular limits, maximum principles, and level domains in the MacLane class

Let A denote the MacLane Class, that is, the class of (nonconstant) analytic functions on the unit disk D having asymptotic limits in a dense subset of the unit circle T. Of central interest here are three desirable properties that a function f∊A may possess: f has angular limits in a metrically dense subset of T f obeys a certain maximum principle, and the components of {|f|>λ} are Jordan regions for each λ∊(inf|f|,sup|f|). It is known, for example, that if f ∊A and f has no infinite asymptotic values, then all of these properties hold. In this note we consider various subclasses of A and give less restrictive conditions which guarantee that the properties are valid. In particular, we show that for the class N of normal functions, relatively weak radial conditions are sufficient.

Subnormality and composition operators on H2

If ϑ is a non-constant analytic function defined on the unit disk D such that ϑ( D ) ⊂ D , the composition operator C ϑ is the operator on the Hardy space H 2 of the unit disk defined by C ϑ f = f ∘ ϑ. In this paper, we investigate the relationship between properties of the symbol ϑ and the subnormality of the operators C ϑ and C ϑ ∗ . The main theorem shows, under a regularity hypothesis, that C ϑ ∗ is subnormal if and only if ϑ is in a restricted class of linear fractional transformations. The two previously known special cases, the affine transformations associated with the Cesaro operator and the automorphisms, are included as the extreme examples. The proof of the main theorem consists of verifying a moment condition and yields, in addition, the construction of a measure μ that exhibits C gj ∗ as multiplication by an analytic function on P 2( μ) and the identification of the minimal normal extension as a weighted sum of shifts.

Functional powers near a fixed point

It is proved that if a function F( z) is analytic in a neighborhood of a fixed point z 0, and if 0 < ¦F′(z 0)¦ < 1 , then there exists a family of related functions F( p, z), each defined in a neighborhood of z 0, which act as functional powers of F( z). In particular, F(0, z) = z, F(1, z) = F( z), and F( p, F( q, z)) = F( p + q, z). It is further demonstrated that the family of functions F( p, z) is identical with the set of nonconstant analytic functions with fixed point z 0 which commute with F( z).

Exit times for elliptic diffusions and BMO

In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.

Analytic Functions Sharing Level Curves and Tracts

A level curve for a nonconstant analytic function is a curve on which the function has constant modulus. Analytic functions should be symmetric about their level curves; any behavior on one side should be the reflection of behavior on the other. This symmetry may be realized locally-one straightens out a bit of the level curve with a local change of coordinates and appeals to the Schwarz reflection principle. Here we prove that, in fact, the only obstructions to global symmetry are presented by compositional factors. Removing these factors allows us to straighten out complete level curves and to exploit the perfect symmetry of the functions remaining. We can then establish, in a unified way, numerous old and new results about shared level curves and tracts-including work going back to Valiron and Cartwright and more recent work of Fuchs, Heins, and others. Given a Riemann surface O' and a nontrivial arc r in 4, we denote by Yr the nonconstant meromorphic functions having constant modulus one on F. When 4 is the Riemann sphere P, the plane C, or the unit disc D and F lies in the extended real line RX, we use the notation 3VR. The key theorem of our work is this:

Finite-to-One Open Mappings

The class of finite-to-one open mappings on manifolds contains some important subclasses. Any non-constant analytic function from a bounded region in its domain of definition is finite-to-one. Church [2] showed that any light strongly open Cn map f: Rn → Rn is discrete. A number of papers concerning discrete open mappings on manifolds have been published; see [1-6; 8-9; 11-14].A result of Černavskiĭ [1] (see also [13]) shows that for any discrete strongly open mapping f : Mn → Nn of an n-manifold into an n-manifold, the branch set of f has dimension less than n – 1. If f is also a closed map, then N(f) is finite and the set of points x for which N(x, f) = N(f) is an open dense connected subset of Mn.

On the radial limits of analytic functions

1. Examples have been given [5, p. 185] of functions f(z), analytic in the unit-circle K: j z j <1, and not identically constant, for which the radial limit f(ei) = lim7r_f(rei) is zero for all e on |z = except for a set of linear measure zero. In view of the Riesz-Nevanlinna theorem [6, p. 197], such functions cannot be bounded, or even of bounded characteristic, in I z I < 1. Functions of this sort appear again whenever we have an analytic function f(z) whose radial limits coincide almost everywhere with the radial limits of a bounded analytic function g(z), for the difference F(z) =f(z) -g(z) has a radial limit zero almost everywhere on I zI = 1. The Riesz-Nevanlinna theorem shows that, if f(z) is bounded, or of bounded characteristic, and if the radial limit values of f(z) coincide almost everywhere on an arc of I z = 1 with the radial limit values of g(z), then F(z) must be identically zero in Iz <1. The object of this note is to discuss certain aspects of the behavior of nonconstant analytic functions whose radial limits vanish almost everywhere on an arc A(01<0<02) Of IzI =1. One result of such a study (which the author plans as a sequel to this note) will be to give some idea of the way in which a function f(z), whose radial limits coincide almost everywhere with the radial limits of a function g(z) of bounded characteristic, can differ from g(z). We shall say that a nonconstant function f(z), analytic in j z j <1, is of class (LP) on an arc A of I zI = 1, if lim, jf(rei) =f(e ) =0 for almost all ei? belonging to the arc A. If the arc A is the whole circumference jzj =1, we shall say simply that the function f(z) is of class (LP). One property of functions which are of class (LP) on an arc A is immediate: the cluster set of f(z) at each point eiGo of A (i.e., the set of all values a with the property that there exists a sequence { z,n }z I <1, lim,0 Zn = eio, such that limn-,f(zn) = a) is the whole plane. For, if there is a point eio on A and a complex number oz which does not belong to the cluster set of f(z) at eiGo, then there is a circular neighborhood V(ei?o) of ei?o such that, in V(efo)qK, the function g(z) = [f(z) -a]-' is analytic and bounded. Since the function g(z) has the constant limit 1/a along almost all normal segments drawn to that arc of I z| =1 which bounds VnK, it follows from a simple corollary of the Riesz-Nevanlinna theorem that g(z), and hence f(z),

On the Existence of Riemannian Manifolds which cannot Carry Non-Constant Analytic or Harmonic Functions in the Small

On the Existence of Riemannian Manifolds which cannot Carry Non-Constant Analytic or Harmonic Functions in the Small