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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20062006:17231

  • Received: 23 January 2006
  • Accepted: 20 April 2006
  • Published:


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.


  • Analytic Function
  • Unit Disk
  • Function Theory
  • Approximation Property
  • Singular Integral


Authors’ Affiliations

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
Department of Mathematics, University of Oradea, Str. Armatei Romane 5, Oradea, 410087, Romania


  1. Alexander JW: Functions which map the interior of the unit circle upon simple regions. Annals of Mathematics. Second Series 1915,17(1):12–22. 10.2307/2007212MathSciNetView ArticleMATHGoogle Scholar
  2. Gal SG: Degree of approximation of continuous functions by some singular integrals. Revue d'Analyse Numérique et de Théorie de l'Approximation 1998,27(2):251–261.MathSciNetMATHGoogle Scholar
  3. Gal SG: Convolution-type integral operators in complex approximation. Computational Methods and Function Theory 2001,1(2):417–432.MathSciNetView ArticleMATHGoogle Scholar
  4. Gal SG: On the Beatson convolution operators in the unit disk. Journal of Analysis 2002, 10: 101–106.MathSciNetMATHGoogle Scholar
  5. Gal SG: Geometric and approximate properties of convolution polynomials in the unit disk. Bulletin of the Institute of Mathematics. Academia Sinica 2006,1(2):307–336.MathSciNetMATHGoogle Scholar
  6. Mocanu PT, Bulboaca T, Salagean GrSt: Geometric Theory of Univalent Functions. Casa Cartii de Stiinta, Cluj; 1999.Google Scholar
  7. Obradović M: Simple sufficient conditions for univalence. Matematichki Vesnik 1997,49(3–4):241–244.MathSciNetMATHGoogle Scholar
  8. Pólya G, Schoenberg IJ: Remarks on de la Vallée Poussin means and convex conformal maps of the circle. Pacific Journal of Mathematics 1958,8(2):295–334.MathSciNetView ArticleMATHGoogle Scholar
  9. Ruscheweyh S, Salinas LC: On the preservation of periodic monotonicity. Constructive Approximation 1992,8(2):129–140. 10.1007/BF01238264MathSciNetView ArticleMATHGoogle Scholar
  10. Stewart I, Tall D: Complex Analysis. Cambridge University Press, Cambridge; 2002.MATHGoogle Scholar
  11. Suffridge TJ: Convolutions of convex functions. Journal of Mathematics and Mechanics 1966, 15: 795–804.MathSciNetMATHGoogle Scholar


© Anastassiou and Gal 2006

This article is published under license to BioMed Central Ltd. 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.