A boundary Schwarz lemma on the classical domain of type <Emphasis Type="Italic">I</Emphasis>

Let R _I (m, n) be the classical domain of type I in ? ^m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R _I (m, n) for a holomorphic self-mapping f of R _I (m, n). We provide a necessary and sufficient condition such that the boundary points of R _I (m, n) are smooth, and give some properties of the smooth boundary points of R _I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R _I (m, n), which may be applied to get some optimal estimates in several complex variables.     

Concave functions,blaschke products,and polygonal mappings

We consider the class Co(p) of all conformal maps of the unit disk onto the exterior of a bounded convex set. We prove that the triangle mappings, i.e., the functions that map the unit disk onto the exterior of a triangle, are among the extreme points of the closed convex hull of Co(p). Moreover, we prove a conjecture on the closed convex hull of Co(p) for all p ∈ (0, 1) which had partially been proved by the authors for some values of p ∈ (0, 1).     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

On the existence of strips inside domains convex in one direction

A plane domain Ω is convex in the positive direction if for every ω ∈ Ω, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and lim_t→∞h^?1(h(z) + t) = 1. We show that if ∠lim_z→?1 Re h(z) = ?∞ and ∠lim_z→?1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.     

On <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-estimates of derivatives of univalent rational functions

We study the growth of the quantity ∫_T|R′(z)|dm(z) for rational functions R of degree n which are bounded and univalent in the unit disk and prove that this quantity can grow like n ^γ , γ > 0, as n → ∞. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result of Dolzhenko, which applies to general (non-univalent) rational functions.     

Characterization of convex domains in ℂ<Superscript>2</Superscript> with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ?ⁿ. Suppose there exists {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ?² with C ² boundary. Suppose that there is a sequence {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

Partial Order Embeddings with Convex Range

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for one of the simplest examples, we prove that every product order embedding σ : ?^? → ?^? with convex range is of the form

$$ \sigma(x)(n)=\left( (x\circ g_{\sigma})+y_{\sigma}\right)(n) ~~~~\text{if}~ n\in K_{\sigma}, $$

(1)

and σ(x)(n) = y _σ (n) otherwise, for all x ∈ ?^?, where K _σ ? ?, g _σ : K _σ → ? is a bijection and y _σ ∈ ?^?. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form ? ^I for some index set I, modulo equality on a comeager subset of the domain, with its ‘natural’ ordering.

     

On some properties of finite sums of ridge functions defined on convex subsets of ℝ<Superscript> <Emphasis Type="Italic">n</Emphasis> </Superscript>

Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets E of the space Rⁿ. It is shown that under some constraints imposed on the summed functions ?_i, in the case when E is open, the continuity of the sum implies the continuity of all ?_i. In the case when E is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions ?_i in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.     

An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions

We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+?+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t ^?β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.     

Order of the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

An exclusive-OR sum of pseudoproducts (ESPP) is a modufo-2 sum of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this sum; the length of a Boolean function in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L ^ESPP(n) on the set of Boolean functions in the class of ESPPs is considered; it is defined as the maximum length of a Boolean function of n variables in the class of ESPPs. It is proved that L ^ESPP(n) = ? (2 ⁿ /n ²). The quantity L ^ESPP(n) also equals the least number l such that any Boolean function of n variables can be represented as a modulo-2 sum of at most l multiaffine functions.     

Bounded projectors and duality in the spaces of functions holomorphic in the unit ball

The paper studies the Banach spaces h _∞(φ), h ₀(φ), and h ¹(η) of harmonic functions over the unit ball in R ⁿ . These spaces depend on a weight function φ and a weight measure η. For a given function φ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures η such that h ¹(η)* ～ h _∞(φ) and h ₀(φ)* ～ h ¹(η).     

An improvement of Ozaki’s P-valent conditions

The old result due to[Ozaki,S.:On the theory of multivalent functions Ⅱ.Sci.Rep.Tokyo Bunrika Daigaku Sect.A,45-87(1941)],says that if f(z) = z~p + ∑_(n=p+1~(a_nz~n))~∞ is analytic in a convex domain D and for some real α we have Re{exp(iα)f~((p))(z)} 0 in D,then f(z) is at most p-valent in ED.In this paper,we consider similar problems in the unit disc B = {z ∈ C:|z| 1}.     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

Conjugacy classes in Möbius groups

Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M _o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M _o (n) depends on the conjugacy of T and T ^?1 in M _o (n). For an element T in M(n), T and T ^?1 are conjugate in M(n), but they may not be conjugate in M _o (n). In the literature, T is called real if T is conjugate in M _o (n) to T ^?1. In this paper we classify real elements in M _o (n). Let T be an element in M _o (n). Corresponding to T there is an associated element T _o in SO(n + 1). If the complex conjugate eigenvalues of T _o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ₁, . . . , θ _k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M _o (n) we have parametrized the conjugacy classes of regular elements in M _o (n). In the parametrization, when T is not conjugate to T ^?1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ? Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ?. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.     

Holomorphic extension of functions along finite families of complex straight lines in an <Emphasis Type="Italic">n</Emphasis>-circular domain

We consider the continuous functions on the boundary of a bounded n-circular domain D in ?ⁿ, n > 1, which admit one-dimensional holomorphic extension along a family of complex straight lines passing through finitely many points of D. The question is addressed of the existence of a holomorphic extension of these functions to D.     

High-order Schwarz-Pick Lemma for the Schur class on the polydisc

In this paper, we give Schwarz-Pick Lemma for functions in the Schur class on the polydisc of ? ⁿ , and generalize some early work of Schwarz-Pick Lemma for functions in the Schur class on the unit disk of C and functions in the Schur-Agler class on the polydisc of ? ⁿ .     

The Cauchy problem for operators with injective symbol in the Lebesgue space <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> in a domain

Let D be a bounded domain in ? ⁿ (n ≥ 2) with infinitely smooth boundary ?D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L ²(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L ²(D) from the Cauchy data given on a relatively open connected set Γ ? ?D and the values Au in D whenever the data belong to L ²(Γ) and L ²(D) respectively.