# Abdullayev F. G.

### Bernstein – Walsh-type polynomial inequalities in domains bounded by piecewise asymptotically conformal curve with nonzero inner angles in the Bergman space

Abdullayev G. A., Abdullayev F. G., Şimşek D.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 583-595

UDC 517.5

We continue the 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 a piecewise asymptotically conformal
curve with nonzero inner angles in the Bergman weight space.

### International conference on the mathematical analysis, differential equations, and their applications (MADEA-8) devoted to the 80th birthday of Academician A. M. Samoilenko

Abdullayev F. G., Samoilenko A. M., Savchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1439-1440

### Polynomial inequalities in regions with interior zero angles in the Bergman space

Abdullayev F. G., Balci S., Imash kyzy M.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 318-336

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

### Application of the Faber polynomials in proving combinatorial identities

Abdullayev F. G., Imash kyzy M., Savchuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 151-164

We study the possibility of application of the 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 expansions of the corresponding conformal mappings. Some examples are presented.

### Polynomial inequalities in quasidisks on weighted Bergman space

Abdullayev G. A., Abdullayev F. G., Tunç E.

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 582-598

We continue studying on the Nikol’skii and Bernstein –Walsh type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman space.

### Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I

Abdullayev F. G., Özkartepe N. P.

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1365-1379

We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue space, where the contour and the weight functions have some singularities.

### On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

Abdullayev F. G., Özkartepe P.

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 579–597

Let *G* ⊂ *ℂ* be a finite region bounded by a Jordan curve *L* := ∂*G*, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \) ), let Δ := {*w* : |*w*| > 1}, and let *w* = Φ(*z*) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let *h*(*z*) be a weight function and let *A* _{ p }(*h,G*), *p* > 0 denote a class of functions *f* analytic in *G* and satisfying the condition $$ {\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }} $$ where *σ* is a two-dimensional Lebesgue measure.

Moreover, let *P* _{ n } (*z*) be an arbitrary algebraic polynomial of degree at most *n* ∈ ℕ. The well-known Bernstein–Walsh lemma states that

In this present work we continue the investigation of estimation (*) in which the norm \( {\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)} \) is replaced by \( {\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0 \) , for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.

### On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

Abdullayev F. G., Özkartepe N. P.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 582-596

Let $\mathbb{C}$ be the complex plane, let $\overline{\mathbb{C}} = \mathbb{C} \bigcup \{\infty\}$, let $G \subset \mathbb{C}$ be a finite Jordan domain with $0 \in G$, let $L := \partial G$, let $\Omega := \overline{\mathbb{C}} \ \overline{G}$, and let $w = \varphi(z)$ be the conformal mapping of $G$ onto a disk $B(0, \rho) := \{w : \; |w | < \rho_0\}$ normalized by $\varphi(0) = = 0,\; \varphi'(0) = 1$, where $\rho_0 = \rho_0 (0, G)$ is the conformal radius of $G$ with respect to 0. Let $\varphi \rho(z) := \int^z_0 [\varphi'(\zeta)]^{2/p}d\zeta$ and let $\pi_{n,p}(z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G, 0)$ that minimizes the integral $\int\int_G|\varphi'(z) - P'_n(z)|^p d \sigma_z$ in the class of all polynomials of degree $\text{deg} P_n \leq n$ such that $P_n(0) = 0$ and $P'_n(0) = 1$. We study the uniform convergence of the generalized Bieberbach polynomials $\pi_{n,p}(z)$ to $\varphi \rho(z)$ on $\overline{G}$ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain better estimates for the rate of convergence in these domains.

### On the Bernstein - Walsh-type lemmas in regions of the complex plane

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 291-302

Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$. Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition $$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$ where $\sigma$ is a two-dimensional Lebesque measure. Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh lemma says that $$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$ Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$ for some regions of complex plane.

### On the Convergence of Fourier Series with Orthogonal Polynomials inside and on the Closure of a Region

Abdullayev F. G., Küçükaslan M.

Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1299-1312

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.

### On Some Properties of Orthogonal Polynomials over an Area in Domains of the Complex Plane. II

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 3-13

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.

### On Some Properties of Orthogonal Polynomials over an Area in Domains of the Complex Plane. I

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1587-1595

We establish conditions for the interference of singularities of a weight function and a contour for orthogonal polynomials over the area of a domain. We obtain new estimates for the rate of growth of these polynomials, which depend on the singularities of the weight function and the contour.