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

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.


Introduction
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: 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: We denote by K(β) the class of convex functions of order β.For simplicity, we also 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 We denote by SS(γ) the class of strongly starlike functions of order γ.We also consider the subclass PS ⊂ A of parabolic starlike functions in U (see [8]), which satisfy the inequality: Recall that an analytic function w of the unit disk U is a Schwarz function if it satisfies the conditions of the Schwarz lemma, i.e., w(0) = 0 and |w(z)| < 1 (z ∈ U).
For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write if there exists a Schwarz function w(z), such that . Furthermore, if the function g is univalent in U, then we have the equivalence: In 1998, Sokó l [11] introduced the class SL ⊂ S * , which consists of the functions In a recent paper, Kargar et al. [2] investigated the class MS(α) defined below, and obtained several radius results for certain well-known function classes.
This paper is organized as follows.In Section 2, we recall certain preliminary lemmas, which are useful in the study of the above classes of functions.In Section 3, we consider some basic properties of the class MS(α), such as integral representation, property of convolution, sufficient condition and coefficient inequalities.In Section 4, we consider radius problems and inclusion relations for certain classes of strongly starlike functions, parabolic starlike functions and SL ⊂ S * , which are closely related to the class MS(α), and the derivations are similar to those used earlier by Sun et al. [13] and Kwon et al. [4].Our results are essential improvements of the corresponding results obtained by Kargar et al. [2].

Preliminaries
Recently, Kargar et al. [2] introduced the analytic function F α and the vertical strip Ω α , which are defined as follows: and where π/2 ≤ α < π.The function F α , defined by (2.1), is convex and univalent in U.
In addition, F α maps U onto Ω α , or onto the convex hull of three points (one of which may be that point at infinity) on the boundary of Ω α .In other words, the image of U may be a vertical strip for π/2 ≤ α < π.In other cases, the image can be for example a half strip, a quadrilateral, or a triangle (see [1]).
We note that the function F α can be written in the form In the recent years, there has been significant interesting results about the class of normalized analytic functions f ∈ A that map U onto vertical strip, see e.g.[3-5, 9, 10, 13, 14].
In order to prove the main results, we need the following lemmas.
Lemma 2. (see [6]) Let h be analytic and convex univalent in U, and Lemma 3. (see [7]) Let the function r(z) given by C n z n be analytic and univalent in U, and suppose that r(z) maps U onto a convex domain.
If the function q(z) given by is analytic in U and satisfies the following subordination relation:

Properties of the class MS(α)
In this section, we will study the properties of the class MS(α).We begin by giving an integral representation for this class.
where w(z) is a Schwarz function.
Next, we give the following property concerning convolutions for the function class MS(α).
We now derive a sufficient condition involving subordination for the functions to be in the class MS(α).
Theorem 3. Let f ∈ A and satisfy the following subordination ) that is, f ∈ MS(α), where F α is given by (2.1).
Proof.For given α (π/2 ≤ α < π), we define the functions q(z) and p(z) by and Then the subordination (2.4) can be written as follows: We note that the function p(z) defined by (3.14) is convex in U and has the form: where B n (α) is given by (2.3).If we let then, by Lemma 3, we see that the subordination (3.15) implies that Then, by equating the coefficients of z n on both sides, we get A simple calculation combined with the inequality (3.16) yields |a 2 | = |A 1 | ≤ 1 and To prove the assertion of Theorem 4, we need to show that We now use mathematical induction to prove (3.17).For n = 3, we have Then suppose that the inequality (3.17) is true for 3 ≤ n ≤ m.We prove the statement for n = m + 1. Straightforward calculations yield which implies that the inequality (3.17) is true for n = m + 1.

Radius problems and inclusion relations
In this section, we first give results on the radius problem involving the function class MS(α).As an application, we obtain inclusion relations for the class MS(α) and the other well-known function classes.The basic method of proof in the following theorem is similar to that used in [13, Theorem 5] (see also [4,Theorem 3.1]). and where Proof.Suppose that f ∈ MS(α).Then, by Lemma 1, the assertion (2.4) holds.Thus, by the definition of subordination, there exists a Schwarz function w(z) such that We put For |z| = r < 1, using the Schwarz lemma, we find that |Q(z) − 1| ≤ e iα − e −iα Q(z) r (|z| = r < 1).(4.6)If we set Q(z) = u + iv, then, upon squaring both sides of (4.6), we get respectively.
We observe that Hence, the origin O lies outside of the disk (4.8), and the disk (4.8) lies in the first and the forth quadrants of uv-plane.
We can obtain the upper and the lower bounds of |Q(z)|: and where N (r, α) > 1 is already given by (4.5).
The following identities are used in the proofs of our main results: and where M 1 (r, α), M 2 (r, α) and N (r, α) are given by (4.3), (4.4) and (4.5), respectively.By using Theorem 5, we derive the following inclusion relations for the class MS(α).
Proof.We first note that Hence the equation P (r) = 0 has a solution in (0, 1).Let r 3 ∈ (0, 1) be the least positive root of H(r) = 0. Then P (r) < 0 for all r < r 3 .Using the same approach as above, we can find r 4 ∈ (0, 1) be the least positive root of the equation (4.16), and the inequality (4.18) holds for all r < r 4 .So if we take r 0 := min{r 3 , r 4 }, then we have f ∈ SL for all z (|z| ≤ r 0 ).
Remark 3. Putting α = π/2 in Theorems 6-8, we obtain the radii of inclusion relations between several known classes and the class MS(α).Furthermore, the results are compared with the corresponding results in [2] (see Table 1).

. 7 )
Thus, Q(z) maps the disk U r = {z : z ∈ C and |z| ≤ r < 1} onto the disk which the center C and radius R are given by