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Initial Maclaurin coefficient estimates for λ -pseudo-starlike bi-univalent functions associated with Sakaguchi-type functions

Abbas Kareem Wanas, Basem Aref Frasin (2022)

Mathematica Bohemica

We introduce and study two certain classes of holomorphic and bi-univalent functions associating λ -pseudo-starlike functions with Sakaguchi-type functions. We determine upper bounds for the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions belonging to these classes. Further we point out certain special cases for our results.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

We consider the class ℋ 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from ℋ 0 and show that if f ∈ 𝒫 H 0 ( α ) and F ∈ 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial...

Integral representations of bounded starlike functions

Frode Rønning (1995)

Annales Polonici Mathematici

For α ≥ 0 let ℱ α denote the class of functions defined for |z| < 1 by integrating 1 / ( 1 - x z ) α if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ℱ 0 . In general, it is only known that bounded starlike functions belong to ℱ α for α > 0.

Integral transforms of functions with restricted derivatives

Johnny E. Brown (2007)

Annales Polonici Mathematici

We show that functions whose derivatives lie in a half-plane are preserved under the Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi integral transforms. We determine precisely how these transforms act on such functions. We prove that if the derivative of a function lies in a convex region then the derivative of its Pommerenke, Chandra-Singh, Libera, Alexander and Bernardi transforms lie in a strictly smaller convex region which can be determined. We also consider iterates of these transforms....

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