Faber polynomial coefficient estimates for analytic bi-close-to-convex functions

Using the Faber polynomials, we obtain coefficient expansions for analytic bi-close-to-convex functions and determine coefficient estimates for such functions. We also demonstrate the unpredictable behavior of the early coefficients of subclasses of bi-univalent functions. A function is said to be bi-univalent in a domain if both the function and its inverse map are univalent there.     

Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative

We introduce and investigate a subclass of analytic and bi-univalent functions defined by a fractional derivative operator in the open unit disk. Using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this class.     

Coefficient estimates for a class of meromorphic bi-univalent functions

Applying the Faber polynomial coefficient expansions to a class of meromorphic bi-univalent functions, we obtain the general coefficient estimates for such functions and also examine their early coefficient bounds. A function univalent in the open unit disk is said to be bi-univalent if its inverse map is also univalent there. Both the technique and the coefficient bounds presented here are new on their own kind. We hope that this article will generate future interest in applying our approach to other related problems.     

Coefficient estimates for certain classes of meromorphic bi-univalent functions

We define a new class of meromorphic bi-univalent functions and use the Faber polynomial expansions to determine the coefficient bounds for such functions. Our results generalize and/or improve some of the previously known results. A meromorphic function is said to be bi-univalent in a given domain Δ if both the function and its inverse map are univalent there.     

Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions

Estimates on the initial coefficients are obtained for normalized analytic functions f in the open unit disk with f and its inverse g=f⁻¹ satisfying the conditions that zf^′(z)/f(z) and zg^′(z)/g(z) are both subordinate to a univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Some radius problems related to a certain subclass of analytic functions

For real parameters α and β such that 0≤α1β,we denote by S(α,β) the class of normalized analytic functions which satisfy the following two-sided inequality:αR(zf′(z)/f(z))β,z∈U,where U denotes the open unit disk.We find a sufficient condition for functions to be in the class S(α,β) and solve several radius problems related to other well-known function classes.     

On optimal polynomial interpolation of analytic functions

This paper discusses the problem of choosing the Lagrange interpolation points T = (t₀, t₁,…, t_n) in the interval −1 t 1 to minimize the norm of the error, considered as an operator from the Hardy space H²(R) of analytic functions to the space C[−1, 1]. It is shown that such optimal choices converge for fixed n, as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R. These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also.     

A special polynomial basis of a space of analytic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 1, pp. 39–42, January–February, 1988.     

Faber polynomial approximation of entire functions of slow growth over Jordan domains

In this paper, we study the $L^p$ -approximation, $2\le p \le \infty $ , of entire functions over Jordan domains by using Faber polynomials. Moreover, the coefficient characterizations of generalized order and generalized type of entire functions for slow growth have been obtained in terms of the $L^p$ -approximation errors. Our results improve the various results of Seremeta (Am Math Soc Transl Ser 2 88:291–301, 1970) and Ganti and Srivastava (Commun Math Anal 7(1):75–93, 2009).     

Coefficient estimates for a certain family of analytic functions involving a q-derivative operator

The Ramanujan Journal - In this paper, we study the coefficient estimates for a class of analytic functions defined by using the q-Ruscheweyh derivative operator. In particular, we investigate the...     

近于凸映照子族全部项齐次展开式的精确估计

本文建立了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照全部项齐次展开式的精确估计.与此同时,作为推论给出了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照精确的增长定理和精确的偏差定理上界估计.所得主要结论表明Cn中单位多圆柱上关于近于凸映照子族和一类近于准凸映照的Bieberbach猜想成立,而且与单复变数的经典结论相一致.     

On the class of bi-univalent functions

In an attempt to answer the question raised by A.W. Goodman, we obtain a covering theorem, a distortion theorem, a growth theorem, the radius of convexity and an argument estimate of f^′(z)$f^{\prime }(z)$ for functions of the class σ of bi-univalent functions.     

关于某些P-叶解析函数类的系数估计

利用拟从属关系引进了一些新的P-叶解析函数的子类,应用解析函数的基本不等式和分析技巧,讨论了相应函数类的系数估计,得到了准确结果,推广了一些相关结果,并给出了Hadamard卷积在Fekete-Szeg问题上的应用.