Abstract
We study a new subclass of functions with symmetric points and derive an equivalent formulation of these functions in term of subordination. Moreover, we find coefficient estimates and discuss characterizations for functions belonging to this new class. We also obtain distortion and growth results. We relate our results with the existing literature of the subject.
1. Introduction and Definitions
Let represent analytic functions in the disc and be defined as:
Let denote “Carathéodory functions” such that and The Möbius function or its rotation acts as an extremal function for the class and maps onto . Recall that consists of functions such that in For , we say that the function is subordinate to and write , if for
For a univalent function , if and only if and For reference, see [1]. Applying subordination, Janowski [2] defined the class for . A function , if
Geometrically, the image lies inside the disk centered on , and diameter ends at and . Clearly, The class is related with the class as: iff, we write
Also , is given by represents various plane curves for the specific values of . Let be the class of complex-valued injective functions and represents the class of starlike functions whereas denotes the class of convex functions. A function is close-to-convex, if and only if there exists a function such that
We denote the class of close-to-convex functions by . This class was introduced by Kaplan in [3]. Sakaguchi (see [4]) defined the class as:
Definition 1. Let Then, , if For , [5] iff , where is the class of convex functions with respect to symmetric points. Various authors studied the class and its subclasses, for detail, see [3, 6–8]. Obviously, it represents the univalent functions. Moreover, it includes the class of convex and odd starlike functions, see [4]. This and other classes are investigated in the literature of the subject; for example, see [9–14].
Definition 2. Let be analytic in defined by (1). We say that , if for we have For more details, see [15]. We see that , where is the class of functions defined in [16]. We study a new class involving .
Definition 3. Let . Then, if for we have where By a simple calculations, we see that (8) is equivalent to
From [16], we have the following lemma.
Lemma 4. For such that if we put where then
Remark 5. Since then Lemma 4 proves that Also from (8), we see that contains close-to-convex functions.
2. Main Results
In the following theorem, we have an equivalent formulation of condition (9) in terms of subordination.
Theorem 6. A function iff for we write where
Proof. Let Then, for we write or where Using subordination, we write because and where Conversely, we assume that (13) holds. Then, there exists with and such that Hence, using , we obtain (9) equivalent to (8), so
Now, we prove sufficient conditions for
Theorem 7. Let be a function given by (10) and If defined by (1) satisfies. where the coefficients are given by (12), then . In particular, if then
Proof. We set for given by (1), where is defined by (10) and have Hence, for , we have the inequality From these calculations, we see that . Also by (20), we can write which is equivalent to (9) and (8). Thus, , and it completes the proof.
The next theorem deals with the coefficient estimates
Theorem 8. Let and Suppose that given by (1) and given by (10) are such that (8) holds. Then, for we have where and is defined by (12). In particular, if then
Proof. If for some , then (9) holds. Using Lemma 4, with , we have
where is an analytic function in , for , and is given by (11). Then
Now, We see that for Thus
Equating coefficients in (27), for , we can also write
where and
Then, we square and integrate along After using the fact , we obtain
Letting , we have
Hence,
Thus, we have the inequality (23) which finishes the proof.
In the following theorem, we prove the growth and distortion theorems for in the class
Theorem 9. If , where and , then where Also, we have where and
Proof. If , then for , (8) holds. It follows from Lemma 4 that in (11) is an odd starlike function. Then For detail, see [17]. From (8), we obtain a function with real part greater then such that It is known, see [18], that Thus, from (36), (38), and (39), we obtain (32). From (32) for , we have This gives us the right-hand side of the inequality (34). To prove the left-hand side of the inequality (34), we must show that it holds for the nearest point from zero, where and Moreover, we have for Since , we know that the function is univalent in the unit disc We conclude that the original image of the line segment This finishes the proof of the inequality (34).
3. Conclusions
In this research, we studied a new subclass of functions with symmetric points and derived an equivalent formulation of these functions in terms of subordination. Moreover, we determined coefficient estimates and discussed characterizations for functions belonging to this new class. We also obtained distortion and growth results. We observed that our findings are related with the existing literature of the subject.
Data Availability
There is no data available.
Disclosure
The research is performed as part of the employment of the “Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan.”
Conflicts of Interest
The authors declare that they have no conflicts of interest.