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.
Highlights
Let us consider the open unit disk D = {z ∈ C; |z| < 1} and A(D) = { f : D → C; f is analytic on D, continuous on D, f (0) = 0, f (0) = 1}
We present some geometric properties of complex Poisson-Cauchy integrals
Concerning the geometric properties of complex Gauss-Weierstrass singular integrals, we present the following
Summary
Let us consider the open unit disk D = {z ∈ C; |z| < 1} and A(D) = { f : D → C; f is analytic on D, continuous on D, f (0) = 0, f (0) = 1}. Therefore, if f ∈ A(D), we have f (z) = z + D. z ∈ D. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 17231, Pages 1–19 DOI 10.1155/JIA/2006/17231
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