F. G. Abdullayev

Mersin University · Department of Mathematics

We study the growth rates of the derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex plane in weighted Lebesgue spaces.

Let $T_n$ be the linear Hadamard convolution operator acting over Hardy space $H^q$, $1\le q\le\infty$. We call $T_n$ a best approximation-preserving operator (BAP operator) if $T_n(e_n)=e_n$, where $e_n(z):=z^n,$ and if $\|T_n(f)\|_q\le E_n(f)_q$ for all $f\in H^q$, where $E_n(f)_q$ is the best approximation by algebraic polynomials of degree a mo...

In this paper we are going to analyze the followingdifference equation $$x_{n+1}=\frac{x_{n-7}}{1+x_{n-1}x_{n-3}x_{n-5}} \quad n=0,1,2, \dots,$$ where $x_{-7}, x_{-6}, x_{-5}, x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_{0} \in \left(0,\infty\right)$.

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.

In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of th...

We obtain exact Jackson-type inequalities in terms of the best approximations and averaged values of the generalized moduli of smoothness in the spaces Sp. For classes of periodic functions defined by certain conditions imposed on the average values of the generalized moduli of smoothness, we determine the exact values of the Kolmogorov, Bernstein,...

In this paper, we study Bernstein-Walsh type estimates for the higher-order derivatives of an arbitrary algebraic polynomial on quasidisks.

We study Bernstein-type and Nikol’skii-type estimates for an arbitrary algebraic polynomial in regions of the complex plane.

УДК 517.5 Отримано точні нерівності типу Джексона в термінах найкращих наближень функцій та усереднених значень їх узагальнених модулів гладкості в просторах . Знайдено точні значення колмогоровських, бернштейнівських, лінійних та проєктивних поперечників в просторах класів періодичних функцій, визначених деякими умовами на усереднені значення їх у...

Let ϕ = {ϕ k } ∞ k=−∞ denote the extended Takenaka-Malmquist system on unit circle T and let σn,ϕ(f), f ∈ L 1 (T), be the Fejér-type operator based on ϕ, introduced by V. N. Rusak. We give the convergence criteria for σn,ϕ(f) in Banach space X(T) := L p (T) ∨ C(T), p ≥ 1. Also, we prove the Voronovskaya-type theorem for σn,ϕ(f) on the class of holo...

Let $\varphi=\{\varphi_k\}_{k=-\infty}^\infty$ denote the extended Takenaka--Malmquist system on unit circle $\mathbb T$ and let $\sigma_{n,\varphi}(f),$ $f\in L^1(\mathbb T)$, be the Fej\'er-type operator based on $\varphi$, introduced by V. N. Rusak. We give the convergence criteria for $\sigma_{n,\varphi}(f)$ in Banach space $X(\mathbb T):=L^p(\...

We describe the set of meromorphic univalent functions in the class Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}, for which the sequence of...

In this paper, we study the uniform convergence of p-Bieberbach polynomials in regions with a finite number of both interior and exterior zero angles at the boundary.

We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class H𝛼 by biharmonic Poisson integrals in the uniform metric.

In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functio...

Exact Jackson-type inequalities are obtained in terms of best approximations and averaged values of generalized moduli of smoothness in the spaces ${\mathcal S}^p$. The values of Kolmogorov, Bernstein, linear, and projective widths in the spaces ${\mathcal S}^p$ are found for classes of periodic functions defined by certain conditions on the averag...

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

In this paper, solution of the following difference equation is examined xn+1=xn−131+xn−1xn−3xn−5xn−7xn−9xn−11, where the initial conditions are positive real numbers.

In the recent years, there has been a lot of interest in studying the global behavior of, the socalled, max-type difference equations; see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The study of max type difference equations has also attracted some attention recently. We study the behaviour of the solutions of the fo...

In weighted Orlicz type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the bes...

In the Orlicz type spaces 𝓢 M , we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre K -functionals in the spaces 𝓢 M .

We determine the subspaces of solutions of the systems of Laplace and heat-conduction differential equations isometric to the corresponding spaces of real functions defined on the set of real numbers.

We continue our investigation of the order of growth of the modulus of an arbitrary algebraic polynomial in the Bergman weight space, where the contour and weight functions have certain singularities. In particular, we deduce a Bernstein–Walsh-type pointwise estimate for algebraic polynomials in unbounded domains with piecewise asymptotically confo...

We study the uniform and mean convergence of the generalized Bieberbach polynomials in regions having a finite number of interior and exterior zero zero angles.

Exact estimates for Faber polynomials, norm of Faber operator, and L1-norm of the generating function for the sequence of Faber operator of univalent functions of class Σ are found. The description of continua of the complex plane for which the norm of Faber operator take the value 1 or 3 is obtained.

In weighted Orlicz type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a ce...

We compare the best approximations of holomorphic functions in the Hardy space H1 by algebraic polynomials and trigonometric polynomials. Particulary, we establish a class of functions f 2 H1 for which the best trigonometric approximation do not coincide with the best algebraic approximation.

In the Orlicz type spaces ${\mathcal S}_{M}$, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre $K$-functionals in the spaces ${\mathcal S}_{M}$.

We describe the set of holomorphic functions from the Hardy space H^q , 1 ≤ q ≤ ∞, for which the best polynomial approximation E n (f) q is equal to | f (n) (0)|/n!.

In the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces S p. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces S p .

In this study, we give some estimates on the Nikolskii-type inequalities for complex algebraic polynomials in regions with piecewise smooth curves having exterior and interior zero angles.

We study the order of growth of the moduli of arbitrary algebraic polynomials in the weighted Bergman space Ap(G, h), p > 0, in regions with zero interior angles at finitely many boundary points. We obtain estimates for algebraic polynomials in bounded regions with piecewise smooth boundary.

We continue the study of estimates of algebraic polynomials in regions bounded by a piecewise asymptotically conformal curve with interior non-zero angles in the weighted Bergman space.

We study the possibility of application of Faber polynomials in proving some combinatorial identities. It is shown that the coefficients of Faber polynomials of mutually inverse conformal mappings generate a pair of mutually invertible relations. We prove two identities relating the coefficients of Faber polynomials and the coefficients of Laurent...

We have obtained the pointwise Bernstein–Walsh type estimation for algebraic polynomials in the unbounded regions with piecewise asymptotically conformal boundary, having exterior and interior zero angles, in the weighted Lebesgue space.

where x−(k+1); x−k; : : : ; x−1; x0 𝜖 (0;∞) and k = 0; 1; 2; : : : .

In this present work, we study the Nikolskii type estimations for algebraic polynomials in the bounded regions with piecewise-asymptotically conformal curve, having interior and exterior zero angles, in the weighted Lebesgue space

This paper is an introduction to soft cone metric spaces. We first define the concept of soft cone metric via soft elements and give basic properties of its. Then, we investigate soft convergence in soft cone metric spaces and prove some important fixed point theorems for contractive mappings on soft cone metric spaces.

We continue our investigation of the Nikol’skii and Bernstein–Walsh-type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman spaces.

In this work, we investigate the order of growth of the modulus of an arbitrary algebraic polynomials in the weighted Lebesgue space, where the contour and the weight functions have some singularities. In particular, we obtain new exact estimations for the growth of the modulus of orthogonal polynomials.

In this paper a solution of the following difference equation xn+1=xn−111+xn−2xn−5xn−8 was investigated, where x−11, x−10, …, x−2, x−1, x0 ∈ (0, ∞).

We study the order of height for the modulus of arbitrary algebraic polynomials with respect to weighted Lebesgue spaces in which the contour and weight functions have certain singularities.

where x−(4k+3), x−(4k+2), . . . , x−1, x0 ∈ (0, ∞) and k = 0, 1, . . . , is studied.

In this present work, we continue studying the Nikol’skii and Bernstein–Walsh type estimations for complex algebraic polynomials in the bounded and unbounded regions bounded by asymptotically conformal curve.

In the present work, we continue to study the growth of the orthogonal polynomials over a contour with a weight function in the weighted Lebesgue space, when the contour and the weight function have some singularities. The case where there is no interference of a weight function and a contour is studied. We consider a piecewise smooth contour with...

In this work, we investigate the order of the growth of the modulus of orthogonal polynomials over a contour and also arbitrary algebraic polynomials in regions with corners in a weighted Lebesgue space, where the singularities of contour and the weight functions satisfy some condition.

In this present work, we continue studying the Bernstein-Walsh type estimations for complex algebraic polynomials in the bounded and unbounded regions with quasiconformal boundary.

We study estimation of the modulus of algebraic polynomials in the bounded and unbounded regions with piecewise-quasismooth boundary, having interior and exterior zero angles, in the weighted Lebesgue space.

In this work, we study the estimation of the modulus of algebraic polynomials in the bounded and unbounded regions with piecewise-quasicircle in the weighted Lebesgue space.

In this paper, we study the estimation for algebraic polynomials in the bounded and unbounded regions bounded by piecewise Dini smooth curve having interior and exterior zero angles.

Let G ⊂ ℂ be a simply connected domain whose boundary L := ∂G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r 0) := {w : |w| < r 0}, satisfying φ(0) = 0, φ′(0) = 1. We consider the following extremal problem for p > 0: $\iint_G {|\phi '(z) - P'_n (z)|^p d\sigma _z \to \min }$ in the class of all polynomi...

Let G subset of C be a finite region bounded by a Jordan curve L := partial derivative G, let Omega := ext (G) over bar (with respect to (C) over bar), let Delta := {w : vertical bar w vertical bar > 1}, and let w = Phi(z) be the univalent conformal mapping of Omega onto Delta normalized by Phi(infinity) = infinity, Phi'(infinity) > 0. Also let h(z...

We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.

In this paper we continue to study two-dimensional analogues of Bernstein-Walsh estimates for arbitrary Jordan domains. MSC: 30A10, 30C10, 41A17.

Let ℂ be the complex plane, let \( \bar{\mathbb{C}}=\mathbb{C}\cup \left\{ \infty \right\} \), let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω := \( \bar{\mathbb{C}}\backslash \bar{G} \), and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ 0) := \( \left\{ {w:\left| w \right|<{\rho_0}} \right\} \) normalized by the co...

Let $A\subseteq\mathbb C$ be a starlike set with a center $a$. We prove that every tangent space to $A$ at the point $a$ is isometric to the smallest closed cone, with the vertex $a$, which includes $A$. A partial converse to this result is obtained. The tangent space to convex sets is also discussed.

We find necessary and sufficient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a ∈ X. Applying this general result we show that each logarithmic spiral has a unique pretangent space at the asymptotic point. Unbounded multiplicative subgroups of C * = C \ {0} having unique pretangent spaces at zero ar...

Let \( G \subset {\mathbb C} \) be a finite region bounded by a Jordan curve \( L: = \partial G \), let \( \Omega : = {\text{ext}}\bar{G} \) (with respect to \( {\overline {\mathbb C}} \)), \( \Delta : = \left\{ {z:\left| z \right| > 1} \right\} \), and let \( w = \Phi (z) \) be a univalent conformal mapping of Ω onto Δ normalized by \( \Phi \left(...

Uniform convergence of the p-Bieberbach polynomials is proved in the case of a simply connected region bounded by a piecewise quasiconformal curve with certain interior zero angles on the corner where two arcs meet.

We study the approximation properties of the extremal polynomials in A p-norm and C-norm. We prove estimates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps.

We describe metric spaces with bounded pretangent spaces and characterize proper metric spaces with proper tangent spaces. We also present the necessary and sufficient conditions under which a tangent space is compact and build a compact ultrametric space X such that some pretangent space to X has the density c.

The aim of this paper is to investigate approximation properties of some extremal polynomials in Ap1, p>0 space. We are interested in finding approximation rate of extremal polynomials to Riemann function in Ap1 and C-norms on domains bounded by piecewise analytic curve.

We find necessary and sufficient conditions under which an arbitrary metric space $X$ has a unique pretangent space at the marked point $a\in X$. Key words: Metric spaces; Tangent spaces to metric spaces; Uniqueness of tangent metric spaces; Tangent space to the Cantor set.

The order of the weight of orthogonal polynomials is analyzed, when this weight function shows singularities on the boundary of a region in the complex plane.

Let G ‰ C be a simply connected region whose boundary L := @G is a Jordan curve and z0 2 G be an arbitrary flxed point. Let w = '(z) be the conformal mapping of G onto the disk D(0;r0) := fw : jwj < r0g; satisfying '(z0) = 0 , ' 0 (z0) = 1. Let us consider the following extremal problem: (1) n(z0) = 1. There exists a polynomial ƒn;p(z) furnishing t...

Let $G\subset C $ be a finite Jordan domain, $z_{0}\in G;$ $B\Subset G$ be an arbitrary closed disk with $z_{0}\in B,$ and $w=\varphi (z,z_{0})$ be the conformal mapping of $G$ onto a disk $\{w:\left| w\right| <r\}$ normalized by $\varphi (z_{0},z_{0})=0$, $\varphi ^{\prime }(z_{0},z_{0})=1$ . It is well known that the Bieberbach polynomials $\{\pi...

GENERAL BIEBERBACH POLYNOMIALS For a finite Borel measure µ and a point λ ∈ C we define the analogue of the Bieberbach polynomial π n as a solution of a suitable extremal problem. We prove a L 2 (µ) convergence of the sequence {π n } to a functioñfunctioñ ϕ and find a characteristic extremal property of˜ϕof˜ of˜ϕ. The conditions for˜ϕfor˜ for˜ϕ = 0...

Let G⊂C be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γj, such that {zj}≔Γj-1∩Γj≠∅, j=1,…,l, where Γ0≔Γl. Denote by αjπ, 0αj⩽2, the angles at zj's between the curves Γj-1 and Γj, exterior with respect to G. Let Φ be a conformal mapping of the exterior C⧹G¯ of G¯=G∪∂G onto the exterior of the unit disk, normed by Φ′(∞)>0. We a...

Let μ be a compactly supported finite Borel measure in ℂ, and let Πn be the space of holomorphic polynomials of degree at most n furnished with the norm of L 2(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Πn their values at a point z ∈ ℂ. The main results demonstrate ho...

Suppose that G is a bounded simply connected domain on the plane with boundary Í\subseteq òln( \tfracdma dw ) dw = - ¥\int {\ln \left( {\tfrac{{d\mu _a }}{{d\omega }}} \right)} d\omega = - \infty equivalent to the completeness of the polynomials in Lt() or to the unboundedness of the calculating functional p p (z0), where p is a polynomial in Lt...

In this work the order of the height of orthogonal polynomials over the region with respect to the weight is analized, when the boundary contour and the weight functions have some singularities.

In this work the estimation of the maximum norm of orthogonal polynomials over the region with respect to the weight is analized.It is observed that the norm of polynomials does not change for the conditions of weight and boundary curve.

We study the rate of convergence of Fourier series of orthogonal polynomials over an area inside and on the closure of regions of the complex plane.

We investigate polynomials that are orthonormal with weight over the area of a domain with quasiconformal boundary. We obtain new exact estimates for the growth rate of these polynomials.

Let C be the extended complex plane; G B(0;r0 ): = { w:|w| < r0 }B(0;\varrho _0 ): = \{ w:|w| < \varrho _0 \} normalized by j(0) = 0 \textand j¢(0) = 1\varphi (0) = 0 {\text{and}} \varphi '(0) = 1 . Let us set $\varphi _p (z): = \int_0^z {\left[ {\varphi '(\zeta )} \right]} ^{{\raise0.7ex\hbox{$\varphi _p (z): = \int_0^z {\left[ {\varphi '(\zeta...

Let GC be a finite Jordan domain, 0G and w = (z) be the conformal mapping of G onto a disc normalized by . It is well known that the uniform convergence of Bieberbach polynomials for the pair (G, 0) to (z) in [Gbar] is governed by the properties of ∂G. In this study, the decrease of to zero and the estimation of this error in domains with interior...

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let jp ( z ): = òx0 x [ f( z) ]2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iintc | jp ( z ) - Px1 (z) |p d0x \iint\limits_c {\left| {\varphi _p \...