On a class of extremal problems

In the paper we introduce a class of trigonometrical polynomial extremal problems depending on a continuous parameter 0≤r≤1. It turns out that the two border cases r=0 and r=1 are known problems investigated earlier by Kamae, Mendes-France, Ruzsa and the present author. We also introduce another set of extremal problems for measures with similar parametrization, and prove a duality relationship between the two type of extremal quantities. The proof relies on a minimax theorem proved earlier by the author. The known duality results are proved as corollaries. 1980 MS Classification. Primary 42A05; Secondary 46B25, 46N05.     

Decompositions into nonoverlapping domains and extremal properties of univalent functions

We apply the method of extremal metrics and certain symmetrization approaches to study problems on conformal mappings of a disk and circular annulus. For instance, we solve the problem on the maximal conformal module in the family of all doubly-connected domains of the form \(E₁∪E₂) with E₁∩E₂=0, r₁E₁, 0≤r₁·r₁≤∞, and diam E₂∩{z:|z|<1}. This generalizes the classical result by A. Mori. We also give a new solution of a problem by P. M. Tamrazov, which was initially solved by V. A. Shlyk. Some new theorems on the covering of a regular system of n rays are obtained for certain classes of convex mappings. Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 139–163. Translated by A. Yu. Solynin.     

On conformal capacity and Teichmüller’s modulus problem in space

We solve an extremal problem for the conformal capacity of certain space condensers. The extremal condenser is conformally equivalent to Teichmüller’s ring. As an application, we give a dimension-free estimate for the minimal conformal capacity of the condensers with platesE, F such thata, b ∈ E,c, d ∈ F, wherea, b, c, d are given points in

Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

Description of sequences of zeros of one class of functions analytic in a half-plane

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition |ƒ(z)| ≤ 0(1) exp (σ| z |η(| z |)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1169–1176, September, 1998.     

The region of values of the system {ƒ(z1), …, ƒ(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Distance sets of well-distributed planar sets for polygonal norms

LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN ^3/2-ε. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

Some approximation theorems

The general theme of this note is illustrated by the following theorem:Theorem 1. Suppose K is a compact set in the complex plane and 0belongs to the boundary ∂K. Let A(K) denote the space of all functions f on K such that f is holo morphic in a neighborhood of K and f(0) = 0.Also for any givenpositive integer m, let A(m, K) denote the space of all f such that f is holomorphic in a neighborhood of K and f(0) =f′(0) = ... =f ^(m)(0) = 0.Then A(m, K) is dense in A(K) under the supre mum norm on K provided that there exists a sector W = re ^iθ; 0≤r≤ δ,α≤ θ≤ β such that W ∩ K = 0. (This is the well- known Poincare’s external cone condition). We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic. Dedicated to Prof. Ashoke Roy on his 62nd birthday     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

α -正规函数的一些积分准则

对 0<α <∞, N^α 是开单位圆盘 D 上满足 sup _{z∈ D}(1-|z|²)^α f^# (z)<∞ 的亚纯函数类, 其中 f^# (z)=|f'(z)|/(1+|f(z)|²) 是 f 的球面导数. 该文给出了 D 上的 α -正规函数类的一些积分准则, 并用 Bergman-Carleson 测度的形式予以表示.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalized weights W = (W_j) and V = (V_n,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {A_j, j ≤ k} the probability V_n,k , where |A_j| is the number of elements of A_j. We impose constraints on the weights by assuming that the resulting random partitions Π _n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Π_n as n tends to infinity. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102.     

Coefficient multipliers of mixed norm space in the ball

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (H _p,q (φ₁), H _u,v (φ₂)) for the values of p, q, u, v in three cases: (i) 0 < p ≤ u ≤ ∞, 0 < q ≤ min(1, v) ≤ ∞. (ii) v = ∞, 0 < p ≤ u ≤ ∞, 1 ≤ u, q ≤ ∞. (iii) 1 ≤ v ≤ 2 ≤ q ≤ ∞, and 0 < p ≤ u ≤ ∞ or 1 ≤ p, u ≤ ∞. The first case extends the result of Blasco, Jevtić, and Pavlović in one variable. The third case generalizes partly the results of Jevtić, Jovanović, and Wojtaszczyk to higher dimensions. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

A remark on the coefficients of bounded polynomials

Let be such that |p(e^iq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |a_v|+|a_n|, 0≤u<v≤n is sharpened.     

Convex hull and extreme points of two families of univalent functions

Closed convex hull and extreme points are obtained for the classR (α) of univalent functions ? inU = {z:¦z ¦< 1}. satisfying inU the conditions ?(0) = 0, ?′(0) = 1Re {f(z)/z} >α, 0 ≤ α < 1. We also obtain the closed convex hull for the classR (α, β) of univalent functionsg inU, satisfying inU the conditionsg (0) = 0,g (0) = 1 and Re{g(z)/F(z)} > β, 0 ? β< 1, wheref ∈ R (α). Integral representations are given for the hulls of these two classes in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example the upper bounds are determined for the coefficients of a function subordinate to some function in R(α) when 1/2 ≤ α < 1.