On a subclass of starlike functions associated with a vertical strip domain

Yong Sun,Zhi-Gang Wang,Antti Rasila,Janusz Sokół

doi:10.1186/s13660-019-1988-8

Abstract

In this paper, we consider a subclass of starlike functions associated with a vertical strip domain. We obtain several results concerned with integral representations, convolutions, and coefficient inequalities for functions belonging to this class. Furthermore, we consider radius problems and inclusion relations involving certain classes of strongly starlike functions, parabolic starlike functions, and other types of starlike functions. The results are essential improvements of the corresponding results obtained by Kargar et al., and the derivations are similar to those used earlier by Sun et al. and Kwon et al.

Highlights

Let A denote the class of the functions of the form ∞f (z) = z + anzn, n=2 (1.1)which are analytic and univalent in the open unit disk U = {z ∈ C : |z| < 1}
We denote by S∗(β) the class of starlike functions of order β
A function f ∈ A is said to be strongly starlike of order γ (0 ≤ γ < 1) if zf (z) arg

Summary

Let A denote the class of the functions of the form

Which are analytic and univalent in the open unit disk U = {z ∈ C : |z| < 1}. A function f ∈ A is said to be starlike of order β (0 ≤ β < 1) if it satisfies the condition zf (z) > β (z ∈ U). We denote by S∗(β) the class of starlike functions of order β. A function f ∈ A is said to be convex of order β (0 ≤ β < 1) if it satisfies the condition zf (z) 1+. We denote by K(β) the class of convex functions of order β. We use the notations S∗ := S∗(0) and K := K(0). A function f ∈ A is said to be strongly starlike of order γ (0 ≤ γ < 1) if zf (z) arg

Sun et al Journal of Inequalities and Applications

Note that the function Fα can be written in the form

We observe that

We now define the continuous function

Inclusion relations