×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Alexander, JW, Functions which map the interior of the unit circle upon simple regions, Annals of Mathematics. Second Series, 17, 12-22, (1915) · 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, 27, 251-261, (1998) · Zbl 1007.41009
[3] Gal, SG, Convolution-type integral operators in complex approximation, Computational Methods and Function Theory, 1, 417-432, (2001) · Zbl 1011.30031
[4] Gal, SG, On the beatson convolution operators in the unit disk, Journal of Analysis, 10, 101-106, (2002) · 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, 1, 307-336, (2006) · 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, 49, 241-244, (1997) · 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, 8, 295-334, (1958) · Zbl 0084.27901
[9] Ruscheweyh, S; Salinas, LC, On the preservation of periodic monotonicity, Constructive Approximation, 8, 129-140, (1992) · 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, 15, 795-804, (1966) · Zbl 0154.32802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.