IJAAMM Int. J. Adv. Appl. Math. and Mech. 6(3) (2019) 27 – 34 (ISSN: 2347-2529)
Journal homepage: www.ijaamm.com
International Journal of Advances in Applied Mathematics and Mechanics
Differential subordination results for Abbas-Starlike function in the
upper half-plane
Research Article
Abbas Kareem Wanas
*
Department of Mathematics,College of Science, University of Al-Qadisiyah, Iraq
Received 10 January 2019; accepted (in revised version) 02 March 2019
Abstract: In the present paper, we define new class of analytic functions in the upper half-plane D = {z ∈ C : Re (z ) > 0}. Also, by
investigating appropriate classes of admissible functions, we obtain differential subordination results for functions
belongs to this new class.
MSC: 130C45 • 30C80
Keywords: Differential subordination • Abbas-starlike functions • Upper half-plane • Admissible functions
© 2019 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
Let H = H (D) stands for the class of functions f : D -→ C which are analytic in the upper half-plane D =
{z ∈ C : Re (z ) > 0} that have the hydrodynamic normalization (see [1, 2, 10])
lim
D3z →∞
( f (z ) - z ) = 0.
Stankiewicz [10] introduced two classes are the class S
*
(D) of starlike functions and the class K (D) of convex func-
tions as follows:
S
*
(D) =
‰
f ∈ H (D): Re
‰
f
0
(z )
f (z )
< 0, z ∈ D
and
K (D) =
‰
f ∈ H (D): Re
‰
f
00
(z )
f
0
(z )
> 0, z ∈ D
.
Next, we define new class for f ∈ H (D) as follows:
Definition 1.1.
A function f ∈ H (D) is said to be Abbas-starlike function, if it satisfies the geometric condition:
Re
( (
f
0
(z )
)
γ
f (z )
)
< 0, γ ≥ 1, z ∈ D.
We denote by A
γ
the family of all Abbas-starlike functions in D. It is observed that for γ = 1, we have the class of
starlike functions in D.
*
E-mail address: abbas.kareem.w@qu.edu.iq.