In this note, we use the notions of -symmetrical, generalized Janowski-type and spirallike functions to define a new class defined in the open unit disk. In particular, we obtain a structural formula, a representation theorem, Marx-Strohacker inequality. Our results continue to hold the covering and distortion properties.

1. Introduction

Let denote the class of analytic functions in the open unit disk . Also, indicates the subclass of which has the form

The family of functions that are univalent in is represented by .

The family of Schwarz functions is denoted by where

Let be analytic, the function is said to be subordinate to in if there exists and , and we denote this by . Whereover is univalent in , then the subordination is equivalent to and .

By using the concept of the subordination, let define the well-known Carathodory class ; , and for any function is said to belong to the class has the representation , for some .

In [1], Janowski introduced the class with , a function analytic in with is said to belong to the class which has the representation .

The class of generalized Janowski functions was introduced in [2]. For arbitrary numbers and with , a function analytic in with is said to belong to the class if and only if

In order to define new classes of symmetrical functions defined in , we first recall the notion of -fold symmetric functions defined in -fold symmetric domain, where is any positive integer. A domain is said to be -fold symmetric if a rotation of about the origin through an angle carries onto itself. A function is said to be -fold symmetric in if for every in we have

The family of all -fold symmetric functions is denoted by ; we get the class of odd univalent functions for . In 1995, Liczberski and Polubinski [3] constructed the theory of -symmetrical functions for and . If is -fold symmetric domain and any integer, then a function is called -symmetrical if for each , We note that the -symmetrical functions are a generalization of the notions of even, odd, and -symmetrical functions.

In [3], we observe that the theory of -symmetrical functions has many interesting applications; we now investigate some results in the classes of analytic functions.

Denote the family of all -symmetrical functions by . We observe that, , , and are the classes of even, odd, and -symmetric functions, respectively.

Theorem 1 (see [3], page 16). For every mapping and a -fold symmetric set , there exists exactly one sequence of -symmetrical functions such that

Furthermore, we say that is -spirallike if and only if is real and

Recently, see [46] obtained many interesting results for various subclasses of Janowski-type functions by using the concept of -symmetrical functions.

By taking motivation from the above-cited work and using the generalized Janowski functions, -symmetrical functions, and -spirallike, we introduce a new subclass of analytic functions.

Definition 2. A function in is said to belong to the class , if where is defined in (5).

Our defined class generalizes many classes by choosing particular values of the parameters for various choices of , and ; Definition 2 yields several known and new subclasses of , as introduced by the authors in [5]; and motivated by Polatolu et al. [2, 7]; introduced by Latha and Darus [8]; defined by Sakaguchi [9]; these class reduce to well-known class defined by Janowski [1].

Lemma 3 (see [5]). If belongs to the class , then and for some , where are defined by ((5)).

Lemma 4 (see [2]). For is an arbitrary fixed point of and , then the set of the values of is in the closed disc with center at and having the radius , where

Lemma 5 (see [10]). Let , then

2. Main Results

Theorem 6. A function belongs to the class if and only if where

Proof. Suppose that , we have Replacing by in (13), we obtain From (13) and (14), we get By differentiation (13), we have From (5) and (15), we get From (5) and (17), we get Integrating repeatedly, we get the required structural formula which proves the necessity. To prove the sufficiency of (10), suppose that (10) holds with . The function defined by (10) is obviously in with and . The following identity can be verified by differentiation where and are given by (10) and (11), respectively. Also, using (10), we have which shows that in .
From (10), since is the root of unity, we conclude that Using (20), (21), and (22), we arrive at the result thus proving the sufficiency of (10).

Lemma 7. If , then for some , where are defined by (5).

Proof. By using Lemma 3 and similar technique proof in Theorem 6 in [5].

Corollary 8. Marx-Strohacker inequality for the class is

Proof. The proof of this corollary is a simple consequence of Lemma 7. Indeed,

The next covering, starlikeness and distortion theorems for the class hold.

Theorem 9. Let , with and . Then, for some and

Proof. Supposing that , it follows that there exists a function such that Combining the above relation with Lemma 7, we have Integrating the above equation along the line connecting the origin with , we obtain our result.

Theorem 10. The radius of starlikeness of the class is

This radius is sharp because the extremal function is

Proof. Since using Lemma 4, that is Using (33) in (34) and after straightforward calculations, we get where . The above inequalities shows that this theorem is true.

Remark 11. (i)For , we obtain (ii)For , we obtain

We also note that if we give another special values to , and , we obtain radius of starlikeness of the subclass of -spirallike functions.

Corollary 12. If , then where .

Proof. For an arbitrary function , we have Using Lemma 5 and after the straightforward calculations, we get the result.

Theorem 13. For , if , then where .

Proof. For function , according to Lemma 7, we have to distinguish the next two cases. (i)For , then there exists a function such that , and by Lemma 5, for, , we getSince , we have For , we can easily use the fact that and ; we have and from (39), we obtain (ii)If , there exists a function such that , and thereforeSince using a similar computation as in the previous case, we deduce Thus, (43) yields to for , which completes the proof of our theorem.

Data Availability

We are applying defined by 1-Liczberski P, Po lubinski J. On ()-symmetrical functions. Mathematica Bohemca 1995; 120(1): 13-28. doi: 10.21136/MB.1995.125897 and extend the class in 1-Al-Sarari F, Latha S, Bulboaca T. On Janowski functions associated with ()-symmetrical functions. Journal of Taibah University for Science 2019; 13(1): 972-978. doi:10.1080/16583655.2019.1665487 2-Polatoglu Y, Bolcal M, Sen A, Yavuz E. A study on the generalization of Janowski functions in the unit disc, Acta Mathematica. Academiae Paedagogicae Nyregyhziensis. 2006; 22; 27-31. 3-Al-Sarari F, Frasin B, AL-Hawary T, Latha S. A few results on generalized Janowski-type functions associated with ()-symmetrical functions. Acta Universitatis Sapientiae, Mathematica 2016; 8(2): 195-205. doi: 10.1515/ausm-2016-0012 4-Polatoglu Y. Growth and distortion theorem for the Janowski alpha-spirallike functions in the unit disc, Stud. Univ. Babes-Bolyai Math 2012; 57; 255-259.

Conflicts of Interest

The authors declare that they have no conflicts of interest.