Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ?: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² ? r ⁿ ? 3r² ? 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains

Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P _n , n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ? be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P _n 's, the case when ? has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P _n 's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function.     

An extension of an inequality of I. Schur

Denote by π_n the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial has the greatest uniform norm in [?1, 1] among all f ∈ ??_n, where (1) Here we show that this extremal property of persists in the wider class of polynomials f ∈ π_n which vanish at ±1, and for which there exist n ? 1 points separating the zeros of and such that for j = 1, …, n ? 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

U(n) Wigner coefficients,the path sum formula,and invariant G-functions

We prove the path sum formula for computing the U(n) invariant denominator functions associated to stretched U(n) Wigner operators. A family of U(n) invariant polynomials G_[λ]⁽ⁿ⁾ is then defined which generalize the _μG_q⁽ⁿ⁾ polynomials previously studied. The G_[λ]⁽ⁿ⁾ polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the _μG_q⁽ⁿ⁾ polynomials. We also give a direct proof of the important transposition symmetry for the G_[λ]⁽ⁿ⁾ polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U(n) Wigner operators.     

A weak-type inequality for uniformly bounded trigonometric polynomials

This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials T _n is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials T _n ^C = {t ∈ T _n : ‖t‖ ≤ C}.     

On the characteristic polynomial of a special class of graphs and spectra of balanced trees

Let H be a simple graph with n vertices and G be a sequence of n rooted graphs G₁,G₂,…,G_n. Godsil and McKay [C.D. Godsil, B.D. McKay, A new graph product and its spectrum, Bull. Austral. Math. Soc. 18 (1978) 21-28] defined the rooted product H(G), of H by G by identifying the root vertex of G_i with the ith vertex of H, and determined the characteristic polynomial of H(G). In this paper we prove a general result on the determinants of some special matrices and, as a corollary, determine the characteristic polynomials of adjacency and Laplacian matrices of H(G).Rojo and Soto [O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005) 97-117] computed the characteristic polynomials and the spectrum of adjacency and Laplacian matrices of a class of balanced trees. As an application of our results, we obtain their conclusions by a simple method.     

The norm ratio of the polynomials with coefficients as binary sequence

Given a positive integerq, the ratio of the 2q-norm of a polynomial which its coefficients form a binary sequence and its 2-norm arose from telecommunication engineering consists of finding any type of such polynomials having the ratio “small”. In this paper we consider some special types of these polynomials, discuss the sharpest possible upper bound, and prove a result for the ratio. MAIN FACTS: A conjecture over a Rudin-Shapiro polynomialP _n which has degree 2 ⁿ ?1 is that for any integerq, the ratio of its 2 _q norm and its 2 norm is asymptotic to the 2qth root of 2 ^q (q+1)^?1. In other words $||P_n ||_{2q} \sim ||P_n ||_2 \sqrt[{2q}]{{\frac{{2q}}{{q + 1}}}}$ . So far only up toq= 2 has been verified. However if the asymptotic behavior is valid for an evenq, then it is also valid for its next consecutive odd integer.     

Minimum 2-tuple dominating set of permutation graphs

For a fixed positive integer k, a k-tuple dominating set of a graph G=(V,E) is a subset D?V such that every vertex in V is dominated by at least k vertex in D. The k-tuple domination number γ _×k (G) is the minimum size of a k-tuple dominating set of G. The special case when k=1 is the usual domination. The case when k=2 was called double domination or 2-tuple domination. A 2-tuple dominating set D ₂ is said to be minimal if there does not exist any D′?D ₂ such that D′ is a 2-tuple dominating set of G. A 2-tuple dominating set D ₂, denoted by γ _×2(G), is said to be minimum, if it is minimal as well as it gives 2-tuple domination number. In this paper, we present an efficient algorithm to find a minimum 2-tuple dominating set on permutation graphs with n vertices which runs in O(n ²) time.     

Weighted polynomial inequalities

For the weights exp (?|x|^λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP _n can decrease on [?1,1] ifP _n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL ^p spaces.     

Continuous embeddings in harmonic mixed norm spaces on the unit ball in ? n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ? ⁿ are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.     

On Constrained Markov–Nikolskii type Inequalities for k-Absolutely Monotone Polynomials

We consider a classical problem of estimating norms of higher order derivatives of an algebraic polynomial via the norms of the polynomial itself. The corresponding extremal problem for general polynomials in the uniform norm was solved by V. A. Markov. In 1926, Bernstein found the exact constant in the Markov inequality for monotone polynomials. It was shown in [3] that the order of the constants in constrained Markov–Nikolskii inequality for k-absolutely monotone polynomials is the same as in the classical one in case \({0 < p \leqq q \leqq \infty}\) . In this paper, we find the exact order for all values of \({0 < p, q \leqq \infty}\) . It turnes out that for the case q < p the constrained Markov–Nikolskii inequality is significantly better than the unconstrained one.     

Where does the sup norm of a weighted polynomial live?

A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)] ⁿ P _n (x), whereP _n is any polynomial of degree at mostn. The (closed) support ∑ ofw(x) may be bounded or unbounded; of special interest is the case whenw(x) has a nonempty zero setZ. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of ∑ —Z. One main result of this paper states that there is a unique compact set (independent ofn andP _n ) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights [w(x)] ⁿ is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.     

Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

Strong convergence of the modified hybrid steepest-descent methods for general variational inequalities

In this paper, we consider the general variational inequality GVI(F, g, C), whereF andg are mappings from a Hilbert space into itself andC is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type methodu _n+1=(1?α+θ _n+1)Tu _n +αu _n ?θ _n+1 g (Tu _n )?λ _n+1 μF(Tu _n ),n≥0. for solving the general variational inequalities. The sequencex _n is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.     

Characterization of intermediate values of the triangle inequality II

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C _n that satisfy 0 ≤ C _n ≤ Σ _j=1 ⁿ ‖x _j ‖ ? ‖Σ _j=1 ⁿ x _j ‖, x ₁,...,x _n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.     

Basic Polynomial Inequalities on Intervals and Circular Arcs

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szeg?–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

The intersection matrix of Brieskorn singularities

To each isolated singularity of a hypersurface of dimensionn, one associates the local fundamental groupG of the moduli space minus the discriminant locus, and a representation σ:G→Aut(H), whereH is then-homology group, with integer coefficients, of the non singular fibre. Although, in general it is very difficult to determine even a presentation ofG, we show that the image of σ can be computed rather easily, by exploiting some relations in a first aproximate presentation ofG, in the case of Brieskorn polynomials namely, polynomials of the type \(x_0^{a_0 } + \cdot \cdot \cdot + x_0^{a_n } \) . In this way we solve an open problem stated by Brieskorn [1] and Pham [8].