Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 17231, 19 pages
doi:10.1155/JIA/2006/17231

Geometric and approximation properties of some singular integrals in the unit disk

George A. Anastassiou1 and Sorin G. Gal2

 1Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2Department of Mathematics, University of Oradea, Str. Armatei Romane 5, Oradea 410087, Romania
Received 23 January 2006; Revised 19 April 2006; Accepted 20 April 2006

Abstract

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.