Application of Conformal Mappings to Inequalities for Polynomials

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. Bibiography: 13 titles.     

Application of Conformal Mappings to Inequalities for Trigonometric Polynomials

In this paper, we obtain inequalities for trigonometric and algebraic polynomials supplementing and strengthening the classical results going back to papers of S. N. Bernstein and I. I. Privalov. The method of proof is based on the construction of the conformal and univalent mapping from a given trigonometric polynomial and on the application of results of the geometric theory of functions of a complex variable to this mapping.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Reduced Modules and Inequalities for Polynomials

We apply the technique of generalized reduced modules in the proof of some inequalities for polynomials. Various estimates of the module of the derivative are obtained in terms of the module of the polynomial, of its leading coefficient, of the distribution of zeros, or of images of critical points. Bibliography: 9 titles.     

Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

Complexification of Multilinear Mappings and Polynomials

We study various methods of complexifying real normed spaces. We see how the notions of duality and complexification are interchangeable. We obtain estimates for the norms of complexified multilinear mappings and polynomials. We see how polynomials can be complexified without reference to the associated multilinear mappings.     

Reproducing Kernels and Conformal Mappings in

In this paper we look at the theory of reproducing kernels for spaces of functions in a Clifford algebra _{0, n}. A first result is that reproducing kernels of this kind are solutions to a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. In the next sections we restrict our attention to Szegö and Bergman modules of monogenic functions. The transformation property of the Szegö kernel under conformal transformations is proved, and the Szegö and Bergman kernels for the half space are calculated.     

Two-dimensional Inverse Scattering and Conformal Mappings

The problem of reconstructing the cross-section of an acousticallysoft or hard cylindrical scatterer is discussed using conformalmapping techniques. A central feature is to investigate minimal requirements onfar field data to ensure the construction of approximate conformalmappings of the exterior of an unknown scatterer on to the exteriorof the unit disc. The method is illustrated through a seriesof examples.     

微分从属运算与保形映射

In this paper we give a sufficient condition on D^δg(z) such that with thecondition.it gnarantees that in the unit disk E implies Some interesting applions ofthis result are also given.     

多值单调映象的Browder-Hartman-Stampacchia型变分不等式

在Ｂａｎａｃｈ空间中,证明了多值单调映象的Ｂｒｏｗｄｅｒ－Ｈａｒｔｍａｎ－Ｓｔａｍｐａｃｃｈｉａ型变分不等式解的存在性定理,统一并推广了「４,５,６,９」中的主要结果。     

Sharp Inequalities for the Product of Polynomials

Let f₁(z),..., f_m(z) be polynomials with complex coefficients,and let their product be of degree n. For any polynomial, let||f|| be the maximum of |f(z)| on the unit circle. We determineconstants C_m < 2 for which for any n. The inequalities are asymptotically sharp as n .This improves earlier results of Gel'fond and Mahler, who gavethe constants e and 2 respectively. If f₁,..., f_m have realcoefficients, we show that for all m 2 and that this is asymptotically sharp. That is,in the real case, the best constant does not depend upon m form 2.     

Conformal Mappings, Hyperanalyticity and Field Dynamics

Generalization of complex analysis to the case of noncommutative algebras of a quaternion-like type is presented. There exists a correspondence between quaternion-differentiable functions and conformal mappings in Euclidean 4-space. For the algebra of biquaternions differentiability conditions are nonlinear and Lorentz-invariant. Starting from these, a version of algebraic field theory, algebrodynamics is suggested. Solutions of basic equations are obtained, and relations to the Maxwell theory disputed.     

Markov- and Bernstein-Type Inequalities for Polynomials with Restricted Coefficients

The Markov-type inequality is proved for all polynomials of degree at most n with coefficients from {-1,0,1} with an absolute constant c. Here ·[0,1] denotes the supremum norm on [0,1]. The Bernstein-type inequality is shown for every polynomial p of the form The inequality is also proved for every analytic function p on the open unit disk D that satisfies the growth condition      

Simplified Lower Bounds for Polynomials with Algebraic Coefficients

We present a very simple method to prove lower bounds for the nonscalar complexity of polynomials with algebraic coefficients.     

On One Method for Factorization of Algebraic Polynomials

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.     

Inequalities for Entire Functions of Finite Degree and Polynomials

The extremal properties of polynomials and entire functions of finite degree not vanishing in the upper half-plane are studied. The exact inequalities obtained complement and strengthen the results by Genchev, Gardner and Govil, Turan, and Lax. Proofs are based on a univalence condition established by Dubinin. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 174–195.     

三阶可微函数的若干不等式

通过建立与Bullen不等式有关的积分恒等式,利用Halder不等式、Grüss不等式、Chebychev不等式,给出三阶可微函数的一些不等式.