Some classes of regular univalent functions

“…Now, suppose that f is slice-starlike in B(0; 1) and that f ∈ N (B(0; 1)). It follows that the coefficients a k ∈ R, for all k ≥ 2 and then since f (z) in starlike on the open unit disk of the complex plane, by the result in [24] we get that L(f )(z) is starlike in the open unit disk of the complex plane. Taking into account Theorem 5.10, it follows that L(f )(q) is slice starlike in the ball B(0; 1).…”

Section: -45)mentioning

confidence: 71%

On some geometric properties of slice regular functions of a quaternion variable

Gal

González-Cervantes²,

Sabadini

2015

Complex Variables and Elliptic Equations

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Abstract. The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, and a Bieberbach-de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from R 4 to itself. As a tool for subordination we define a suitable notion of composition of slice regular functions which is of independent interest.

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“…Krzyż and Lewandowski [16] constructed an example to show that Λ does not carry S into S. Another familiar integral operator Θ : A → A is the Libera operator [18] defined by…”

Section: Introductionmentioning

confidence: 99%