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On a subclass of starlike functions associated with a vertical strip domain
Journal of Inequalities and Applications volume 2019, Article number: 35 (2019)
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.
1 Introduction
Let \(\mathcal{A}\) denote the class of the functions of the form
which are analytic and univalent in the open unit disk \(\mathbb {U}=\{z\in \mathbb{C}: |z|<1\}\). A function \(f\in\mathcal{A}\) is said to be starlike of order β (\(0\leq\beta<1\)) if it satisfies the condition
We denote by \(\mathcal{S}^{*}(\beta)\) the class of starlike functions of order β. A function \(f\in{\mathcal {A}}\) is said to be convex of order β (\(0\leq\beta<1\)) if it satisfies the condition
We denote by \(\mathcal{K}(\beta)\) the class of convex functions of order β. For simplicity, we also use the notations \(\mathcal{S}^{*}:=\mathcal {S}^{*}(0)\) and \(\mathcal{K}:=\mathcal{K}(0)\).
A function \(f\in\mathcal{A}\) is said to be strongly starlike of order γ (\(0\leq\gamma<1\)) if
We denote by \(\mathcal{SS}(\gamma)\) the class of strongly starlike functions of order γ. We also consider the subclass \(\mathcal {PS}\subset\mathcal{A}\) of parabolic starlike functions in \(\mathbb {U}\) (see [8]), which satisfy the inequality
Recall that an analytic function w in the unit disk \(\mathbb {U}\) is a Schwarz function if it satisfies the conditions of the Schwarz lemma:
For two analytic functions f and g in \({\mathbb {U}}\), we say that the function f is subordinate to g in \({\mathbb {U}}\) and write
if there exists a Schwarz function \(w(z)\) such that
It is well known that if \(f(z)\prec g(z)\) (\(z\in{ \mathbb {U}}\)), then \(f(0)=g(0)\) and \(f({\mathbb {U}})\subset g({\mathbb {U}})\). Furthermore, if the function g is univalent in \({\mathbb {U}}\), then we have the equivalence
In 1998, Sokół [11] introduced the class \(\mathcal {SL}\subset\mathcal{S}^{*}\) consisting of the functions \(f\in \mathcal{A}\) such that
Recently, Kargar et al. [2] investigated the class \(\mathcal{MS}(\alpha)\) (see Definition 1) and obtained several radius results for certain well-known function classes.
Definition 1
A function \(f\in\mathcal{A}\) is said to belong to the class \(\mathcal{MS}(\alpha)\) (\(\pi/2\leq\alpha<\pi\)) if it satisfies the following conditions:
Remark 1
From the inequalities (see [2])
it is clear that
where the class \(\mathcal{S}(\beta, \gamma)\), \(0\leq\beta <1<\gamma\), was considered recently by Kwon et al. [4].
This paper is organized as follows. In Sect. 2, we recall certain preliminary lemmas, which are useful in the study of the mentioned classes of functions. In Sect. 3, we consider some basic properties of the class \(\mathcal {MS}(\alpha)\), such as integral representation, property of convolution, sufficient condition, and coefficient inequalities. In Sect. 4, we consider radius problems and inclusion relations for certain classes of strongly starlike functions, parabolic starlike functions, and \(\mathcal{SL}\subset\mathcal{S}^{*}\), which are closely related to the class \(\mathcal{MS}(\alpha)\), 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].
2 Preliminaries
Recently, Kargar et al. [2] introduced the analytic function \(F_{\alpha}\) and the vertical strip \(\varOmega_{\alpha}\) defined as follows:
and
where \(\pi/2\leq\alpha<\pi\). The function \(F_{\alpha}\) defined by (2.1) is convex and univalent in \(\mathbb {U}\). In addition, \(F_{\alpha}\) maps \(\mathbb {U}\) onto \(\varOmega_{\alpha}\) or onto the convex hull of three points (one of which may be at infinity) on the boundary of \(\varOmega_{\alpha}\). In other words, the image of \(\mathbb {U}\) may be a vertical strip for \(\pi /2\leq\alpha<\pi\). In other cases, the image can be, for example, a half strip, a quadrilateral, or a triangle (see [1]).
Note that the function \(F_{\alpha}\) can be written in the form
where
In the recent years, there has been significant interesting results about the class of normalized analytic functions \(f\in\mathcal{A}\) that map \(\mathbb {U}\) onto vertical strip; see, for example, [3,4,5, 9, 10, 13, 14].
To prove the main results, we need the following lemmas.
Lemma 1
(see [2])
Let \(f\in\mathcal{A}\). Then \(f\in\mathcal{MS}(\alpha)\) (\(\pi /2\leq\alpha<\pi\)) if and only if
Lemma 2
(see [6])
Let h be analytic and convex univalent in \(\mathbb {U}\), and let \(\beta ,\gamma\in\mathbb{R}\) with \(\Re (\beta h(z)+\gamma )\geq0\). If q is analytic in \(\mathbb {U}\) with \(q(0)=h(0)\), then
Lemma 3
(see [7])
Let the function \(r(z)\) given by
be analytic and univalent in \(\mathbb {U}\), and suppose that \(r(z)\) maps \(\mathbb {U}\) onto a convex domain. Suppose that the function
is analytic in \(\mathbb {U}\) and satisfies the following subordination relation:
Then
3 Properties of the class \(\mathcal{MS}(\alpha)\)
In this section, we study the properties of the class \(\mathcal {MS}(\alpha)\). We begin by giving an integral representation for this class.
Theorem 1
A function \(f\in\mathcal{MS}(\alpha)\) (\(\pi/2\leq\alpha<\pi\)) if and only if
where \(w(z)\) is a Schwarz function.
Proof
For \(f\in\mathcal{MS}(\alpha)\), we know from Lemma 1 that (2.4) holds. It follows that
where the Schwarz function \(w(z)\) is analytic in \(\mathbb {U}\) with \(w(0)=0\) and \(|w(z)|<1\) (\(z\in \mathbb {U}\)). We next see from (3.2) that
which, upon integration, yields
Assertion (3.1) of Theorem 1 now follows from (3.3). □
Example 1
Let \(w(z)=z\) in Theorem 1. Then the function \(f_{\alpha }\in\mathcal{MS}(\alpha)\) (\(\pi/2\leq\alpha<\pi\)) is given by
Next, we give the following property concerning convolutions for the function class \(\mathcal{MS}(\alpha)\).
Theorem 2
A function \(f\in\mathcal{MS}(\alpha)\) (\(\pi/2\leq\alpha<\pi\)) if and only if
where ∗ denotes the Hadamard product, \(0<\theta<2\pi\), and \(\theta -\alpha\neq\pi\).
Proof
Assume that \(f\in\mathcal{MS}(\alpha)\). Then, by Lemma 1, we observe that (2.4) holds. This implies that
Condition (3.5) can now be written as follows:
Note that
Thus by (3.6) and (3.7) we obtain assertion (3.4) of Theorem 2. □
We now derive a sufficient condition involving subordination for the functions to be in the class \(\mathcal{MS}(\alpha)\).
Theorem 3
Let \(f\in\mathcal{A}\) satisfy the subordination
Then
that is, \(f\in\mathcal{MS}(\alpha)\), where \(F_{\alpha}\) is given by (2.1).
Proof
Consider the function \(p(z)\) such that
Then
We have
or
Note that
Moreover, by (3.11) and (3.12) in Lemma 2 we have
or by (3.10) we have
Therefore by Lemma 1 we obtain that \(f\in\mathcal {MS}(\alpha)\). □
Remark 2
It is well known that \(\mathcal{K}\subset\mathcal{S}^{*}(1/2)\). In view of (1.3) and Theorem 3, we can obtain \(\mathcal{K} (\varPhi(\alpha) )\subset\mathcal{S}^{*} (\varPhi(\alpha) )\) for \(\pi/2\leq\alpha<\pi\), where \(\varPhi (\alpha)=1+(\alpha-\pi)/(2\sin\alpha)\).
Now, we present bounds for the coefficients of functions of the class \(\mathcal{MS}(\alpha)\). The basic method of proof is similar to that used in [12, Thm. 3.1].
Theorem 4
Let \(f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\in\mathcal{MS}(\alpha)\). Then
Proof
For given α (\(\pi/2\leq\alpha<\pi\)), we define the functions \(q(z)\) and \(p(z)\) by
and
Then the subordination (2.4) can be written as follows:
Note that the function \(p(z)\) defined by (3.14) is convex in \(\mathbb {U}\) and has the form
where \(B_{n}(\alpha)\) is given by (2.3). If we let
then by Lemma 3 we see that the subordination (3.15) implies that
Now (3.13) implies that
Then by equating the coefficients of \(z^{n}\) on both sides we get
A simple calculation combined with inequality (3.16) yields \(|a_{2}|=|A_{1}|\leq1\) and
To prove Theorem 4, we need to show that
We prove (3.17) by induction. For \(n=3\), we have
Then suppose that inequality (3.17) is true for \(3\leq n\leq m\). We prove the statement for \(n=m+1\). Straightforward calculations yield
which implies that inequality (3.17) is true for \(n=m+1\). □
4 Radius problems and inclusion relations
In this section, we first give results on the radius problem involving the function class \(\mathcal{MS}(\alpha)\). As an application, we obtain inclusion relations for the class \(\mathcal{MS}(\alpha)\) and the other well-known function classes. The basic method of the proof in the following theorem is similar to that used in [13, Thm. 5] (see also [4, Thm. 3.1]).
Theorem 5
Let \(f\in\mathcal{MS}(\alpha)\). Then, for each z (\(|z|=r<1\)),
and
where
Proof
Suppose that \(f\in\mathcal{MS}(\alpha)\). Then by Lemma 1 assertion (2.4) holds. Thus by the definition of subordination there exists a Schwarz function \(w(z)\) such that
We put
which readily yields
For \(|z|=r<1\), using the Schwarz lemma,
we find that
If we set \(Q(z)=u+iv\), then upon squaring both sides of (4.6) we get
Thus \(Q(z)\) maps the disk
onto the disk with center C and radius R given by
We observe that
and
Hence the origin O lies outside of the disk (4.8), and the disk (4.8) lies in the first and forth quadrants of the uv-plane.
We can obtain upper and lower bounds of \(|Q(z)|\):
and
where \(N(r,\alpha)>1\) is already given by (4.5).
Furthermore, a simple geometric observation shows that (4.7) implies
where \(M_{1}(r,\alpha)\) and \(M_{2}(r,\alpha)\) are given by (4.3) and (4.4), respectively.
For \(|z|=r<1\), we have
Thus by (4.9)–(4.12) we easily get assertions (4.1) and (4.2) of Theorem 5. □
The following identities are used in the proofs of our main results:
and
where \(M_{1}(r,\alpha)\), \(M_{2}(r,\alpha)\), and \(N(r,\alpha)\) are given by (4.3), (4.4), and (4.5), respectively.
Using Theorem 5, we derive the following inclusion relations for the class \(\mathcal{MS}(\alpha)\).
Theorem 6
Let
Then
where \(r_{1}\in(0,1)\) is the least positive root of the equation
where \(M_{1}(r,\alpha)\), \(M_{2}(r,\alpha)\), and \(N(r,\alpha)\) are given by (4.3), (4.4), and (4.5), respectively.
Proof
We first note that
Hence by Theorem 5, for \(f\in\mathcal{MS}(\alpha)\), we have
Thus, for the function \(f\in\mathcal{SS}(\gamma)\), it suffices to prove the inequality
We now define the continuous function
In view of (4.13) and (4.14), we can show that
Thus, the equation \(G(r)=0\) has a solution in \((0,1)\). Let \(r_{1}\in (0,1)\) be the least positive root of \(G(r)=0\). Then \(G(r)<0\) for all \(r< r_{1}\). Hence f is a strongly starlike function of order γ for z (\(|z|\leq r_{1}\)). □
Theorem 7
Let \(\pi/2\leq\alpha<\pi\). Then
where \(r_{2}\in(0,1)\) is the least positive root of the equation
where \(M_{1}(r,\alpha)\), \(M_{2}(r,\alpha)\), and \(N(r,\alpha)\) are given by (4.3), (4.4), and (4.5), respectively.
Proof
Note that \(f\in\mathcal{PS}\) if and only if the function \(zf'(z)/f(z)\) is in the parabolic region given by
Thus by combining (4.1) and (4.2), for the function \(f\in \mathcal{PS}\) in \(\mathbb {U}\), it suffices to show that
that is,
We now define the continuous function
In view of (4.13) and (4.14), we have
Hence the equation \(H(r)=0\) has a solution in \((0,1)\). Let \(r_{2}\in (0,1)\) be the least positive root of \(H(r)=0\). Then \(H(r)<0\) for all \(r< r_{2}\). Therefore we have \(f\in\mathcal{PS}\) for all z (\(|z|\leq r_{2}\)). □
Theorem 8
Let \(\pi/2\leq\alpha<\pi\). Then
where \(r_{0}:=\min\{r_{3}, r_{4}\}\), and \(r_{3}, r_{4}\in(0,1) \) are the least positive root of the equations
and
respectively, where \(M_{1}(r,\alpha)\), \(M_{2}(r,\alpha)\), and \(N(r,\alpha)\) are given by (4.3), (4.4), and (4.5), respectively.
Proof
Note that \(f\in\mathcal{SL}\) if and only if the function \(zf'(z)/f(z)\) is in the bounded region given by
Thus by combining (4.1) and (4.2), for the function \(f\in \mathcal{SL}\) in \(\mathbb {U}\), it suffices to show that
that is,
and
respectively. We define the continuous function
In view of (4.13) and (4.14), we have
Hence the equation \(P(r)=0\) has a solution in \((0,1)\). Let \(r_{3}\in (0,1)\) be the least positive root of \(H(r)=0\). Then \(P(r)<0\) for all \(r< r_{3}\). Using the same approach as before, we can find \(r_{4}\in(0,1)\) to be the least positive root of equation (4.16), and inequality (4.18) holds for all \(r< r_{4}\). So if we take \(r_{0}:=\min\{r_{3}, r_{4}\}\), then we have \(f\in\mathcal{SL}\) for all z (\(|z|\leq r_{0}\)). □
Remark 3
Putting \(\alpha=\pi/2\) in Theorems 6–8, we obtain the radii of inclusion relations between several known classes and the class \(\mathcal{MS}(\alpha)\). Furthermore, the results are compared with the corresponding results in [2] (see Table 1).
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Funding
The present investigation was supported by the Natural Science Foundation of Hunan Province under Grant no. 2016JJ2036 of the People’s Republic of China.
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Sun, Y., Wang, ZG., Rasila, A. et al. On a subclass of starlike functions associated with a vertical strip domain. J Inequal Appl 2019, 35 (2019). https://doi.org/10.1186/s13660-019-1988-8
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DOI: https://doi.org/10.1186/s13660-019-1988-8