Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: GEOMETRIC AND APPROXIMATION PROPERTIES OF SOME SINGULAR INTEGRALS IN THE UNIT DISK | GEOMETRIC AND APPROXIMATION PROPERTIES OF SOME SINGULAR INTEGRALS IN THE UNIT DISK GEORGE A. ANASTASSIOU AND SORIN G. GAL Received 23 January 2006 Revised 19 April 2006 Accepted 20 April 2006 The purpose of this paper is to prove several results in approximation by complex Picard Poisson-Cauchy and Gauss-Weierstrass singular integrals with Jackson-type rate having the quality of preservation of some properties in geometric function theory like the preservation of coefficients bounds positive real part bounded turn starlikeness and convexity. Also some sufficient conditions for starlikeness and univalence of analytic functions are preserved. Copyright 2006 G. A. Anastassiou and S. G. Gal. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let us consider the open unit disk D z e C Izl 1 and A D f D C f is analytic on D continuous on D f 0 0 f 0 1 . Therefore if f e A D we have f z z zr 2 akZk for all z e D. For f e A D and ỉ e R ỉ 0 let us consider the complex singular integrals Pỉ f z 1 f zéu e-Wỉdu z e D 2S -to . ỉ cn f zeiu . _ . . ỉ c to f ze-iu 1 _ Qỉ f z ỉ f2zsdu z e D Q f z ỉ f du z e D n -n u2 ỉ2 q n -to u2 ỉ2 . 2ỉ3 c to f zeiu . Rỉ f z 2ỉ 2 du z e D n J-to u2 ỉ2 Wỉ f z r f zéu e-u2 ỉdu z e D ỵjnỉ j-n W f z f f ze-iu e-u2 ỉdu z e D. yỊnỉ J-to Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 17231 Pages 1-19 DOI JIA 2006 17231 2 Geometric and approximation properties Here Pị f is said to be of Picard type Qị f Q f and Rị f are said to be of Poisson-Cauchy type and Wị f and Wf f are said to be of Gauss-Weierstrass type. In the very recent papers 3-5 classes of convolution complex polynomials were introduced and their approximation properties regarding rates global smoothness preservation properties and some geometric properties .
