Geometric and approximation properties of some singular integrals in the unit disk. (English) Zbl 1193.42075

Summary: 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.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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[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. 10.2307/2007212 · JFM 45.0672.02
[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. · Zbl 1007.41009
[3] Gal SG: Convolution-type integral operators in complex approximation.Computational Methods and Function Theory 2001,1(2):417-432. · Zbl 1011.30031
[4] Gal SG: On the Beatson convolution operators in the unit disk.Journal of Analysis 2002, 10: 101-106. · Zbl 1173.30322
[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. · Zbl 1109.30032
[6] Mocanu PT, Bulboaca T, Salagean GrSt: Geometric Theory of Univalent Functions. Casa Cartii de Stiinta, Cluj; 1999.
[7] Obradović M: Simple sufficient conditions for univalence.Matematichki Vesnik 1997,49(3-4):241-244. · Zbl 0992.30005
[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. · Zbl 0084.27901
[9] Ruscheweyh S, Salinas LC: On the preservation of periodic monotonicity.Constructive Approximation 1992,8(2):129-140. 10.1007/BF01238264 · Zbl 0746.42003
[10] Stewart I, Tall D: Complex Analysis. Cambridge University Press, Cambridge; 2002. · Zbl 0501.30001
[11] Suffridge TJ: Convolutions of convex functions.Journal of Mathematics and Mechanics 1966, 15: 795-804. · Zbl 0154.32802
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