Uniformly Alpha-Quasi-Convex Functions Defined by Janowski Functions

Abstract

In this work, we aim to introduce and study a new subclass of analytic functions related to the oval and petal type domain. This includes various interesting properties such as integral representation, sufficiency criteria, inclusion results, and the convolution properties for newly introduced class.

1. Introduction and Preliminaries

Let be the class of functions of the formwhich are analytic in the open unit disk . Let , , , and be the subclasses of consisting of functions which are univalent, starlike of order , convex of order , and close-to-convex of order , respectively, . Clearly , the class of starlike univalent functions, , the class of convex univalent functions, and , the class of close-to-convex univalent functions.

For analytic functions and , the convolution or Hadamard product is denoted by and is defined by

A function is said to be subordinate to the function , written symbolically as , if there exists a Schwarz function such that where , for . The class consists of functions which are analytic in with and where . If a function belongs to , then it maps on to the domain , defined by The domain represents an open circular disk, having center at real axis and whose diameter points are and . For and , the class reduces to the well-known class of Carathèodory functions; see [1]. The class is connected with the class of Carathèodory functions by the following relation. Janowski [2] introduced the classes and and studied them comprehensively. He established the Alexander type relation between these classes.

In 1999, Kanas and Wiśniowska [3] gave the concept of general conic domain by introducing a domain , defined by The domain represents a right half plane when , the hyperbolic regions for , a parabolic region for , and it represents elliptic regions when . The extremal function for conic domain is the mapping with and and is defined bywhere , , , and is chosen such that , is Legendre’s complete elliptic integral of first kind, and is complementary integral of ; for more details, see [3]. Meanwhile, Kanas and Wiśniowska [3, 4] introduced a class which maps on to the conic regions, defined by the domain . They also defined the classes and of -uniformly convex functions and their corresponding -starlike functions; see [3, 4]. Recently Noor and Malik [5] introduced new geometrical structures of oval and petal type shape as image domain and defined the class of functions which give these types of mappings. The concepts of Janowski functions and conic domains are combined together to define the class which represents the oval and petal type regions as image domain.

Definition 1 (see [5]). A function if and only if, wherewith as defined in (8).

Graphically, the function takes all the values in the domain , , which is defined as or, equivalently, Noor and Malik [5] also introduced two new classes and related to the domain whose geometry is oval and petal type regions subject to the values of , , and beta. These classes are defined as follows.

Definition 2 (see [5]). A function is said to be in the class ,  ,  , if and only ifor, equivalently,

Definition 3 (see [5]). A function is said to be in the class ,  ,  , if and only if or, equivalently,

Motivated from the recent work presented by Noor and Malik [5], we define the following new subclass of analytic functions associated with domain .

Definition 4. A function is said to be in the class , , , , , if and only ifor wherefor some

It follows from the above definition that if and only ifwhere is the class of uniformly Janowski close-to-convex functions, introduced and studied by Mahmood et al. [6].

Special Cases. By setting certain values to different parameters, the class coincides with several renowned classes of analytic functions, as given below.(1), are well-known classes of uniformly Janowski close-to-convex and uniformly Janowski quasi-convex functions respectively, introduced and studied by Mahmood et al. [6].(2) is the well known class of uniformly close-to-convex functions, introduced and studied by Noor et al. [7].(3) is the well known class of alpha-quasi-convex functions, introduced and studied by Noor and Aboudi [8].(4) is the well known class of alpha Janowski convex functions, introduced and studied by Selvaraj and Thirupathi [9].

Throughout this paper, we assume that , , , and unless otherwise stated.

2. A Set of Lemmas

To prove our main results we need the following Lemmas.

Lemma 5 (see [5]). Let . Then where

Lemma 6 (see [5]). Let with and be given by Then where is defined by (23).

Lemma 7 (see [10]). Let and be in the classes and , respectively. Then for every function analytic in with , we have where denotes the closed convex hull .

Lemma 8 (see [6]). Let and . Then .

3. Main Results

Theorem 9. Let . Then, the function is defined such thatbelongs to the class .

Proof. From (27), we can write and the result follows by using (20).

Theorem 10. A function having the form (1) is in the class if it satisfies

Proof. Assuming that (29) holds true, then it is sufficient to show thatNow considerIn the last inequality, we have used the following coefficient inequality for functions belonging to the class (see [5]):The last inequality is bounded above by 1, ifWhen , Theorem 10 reduces to the following known result, proved in [6].

Corollary 11. If a function satisfiesthen

When , Theorem 10 takes the form of following known result, proved in [6].

Corollary 12. A function is said to be in the class if it satisfies the condition

Theorem 13. Let and be of the form (1). Then for , we have where is given by (22).

Proof. By definition, for , we getwhere We can write (37) asNow using the series expansion, (39) reduces to Equating coefficients of on both sides, one may get This implies thatNow applying Lemmas 5 and 6, (42) reduces towhere is given by (22). This completes the proof.

By setting in Theorem 13, we have the following known result, proved by Mahmood et al. [6].

Corollary 14. Let and be the th coefficient of Taylor series of . Then, for , we obtain where is given by (22).

For , Theorem 13 takes the form of following known result, proved by Mahmood et al. [6].

Corollary 15. Let the function of the form (1) belong to the class . Then, where and is given by (22).

When we consider , , and in Theorem 13, we obtain the following known result, proved by Noor et al. [7].

Corollary 16. Let and be the th coefficient of Taylor series of . Then,

By Putting , , and in Theorem 13, we have the following known result proved by Selvaraj and Thirupathi [9].

Corollary 17. Let and be of the form (1). Then,

When we take , , and in Theorem 13, we obtain the following known result proved by Noor and Aboudi [8].

Corollary 18. Let and be the th coefficient of Taylor series of . Then,

Theorem 19. Let , , and be any convex univalent function in . Then

Proof. Let , . Then by (20), we have Using Lemma 8, one may have which implies that Hence by (20), . This completes the proof.

Applications of Theorem 19

Theorem 20. Let . Then the functions , such that belong to the class .

Proof. The proof follows immediately, when we observe that where is convex in such that

Theorem 21. For , we have .

Proof. For , the result is obvious. So we assume that and using the integral representation for , we havewhere , since , where is a convex function in . Now using Lemma 8, we conclude that . This completes the proof.

Theorem 22. If , Then,

Proof. Let . Then Now consider Since is convex, and consequently . This completes the proof.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge Professor Dr. Salim ur Rehman, V.C. Sarhad University of Science & IT, for providing excellent research and academic environment.