The main object of the present paper is to investigate a number of useful properties such as inclusion relation, distortion bounds, coefficient estimates, and subordination results for a new subclass of analytic functions which are defined here by means of a linear operator. Several known consequences of the results are also pointed out.

1. Introduction and Definitions

Let denote the set of all functions of the formwhich are analytic in the open unit disk . Suppose that and are analytic in ; then we say that is subordinate to , written as or , if there exists a function , which is analytic in with and such that Therefore, implies . Moreover, if the function is univalent in , then the following equivalence holds:Furthermore, let , , denote the family of all functions that are analytic in the open unit disk with and satisfy Geometrically, by (3), a function is in the class , , if and only if and , where circular domain is defined byThe domain , , represents an open circular disk centered on the real axis with diameter end points and with .

With the help of the class , we now define the classes and of Janowski starlike and Janowski convex functions as below: These classes were introduced by Janowski [1]. The extension of Janowski function was discussed by Kuroki, Owa, and Srivastava [2] by choosing the complex parameters and with the following conditions: Later on, Kuroki and Owa [3] discussed the fact that the condition can be omitted from the conditions in part of (7). Janowski functions are being studied and extended in different directions by several renowned mathematicians like Noor and Arif [4], Arif et al. [5], Polatolu [6], Cho [7], Cho et al. [8, 9], Liu and Noor [10], Liu and Patel [11], Liu and Srivastava [12, 13], etc.

For a function of the form (1) and , the Slgean operator [14] is defined by It is easy to see that the series is convergent in the unit disc for each . Further, we have . Also, we consider the following differential operator: for any integers. Then, for given by (1), we know thatUsing this Slgean operator along with the concepts of Janowski functions, we now define a subclass of as follows.

Definition 1. If , then if and only ifwhere , , and .

Special Cases. In literature, various interesting subfamilies of analytic and univalent functions associated with circular domain have been studied from a number of different view points which are closely related with the class

For example, if we set in (11), we get the class defined as and further, by making , , in , we obtain the familiar class of starlike functions with respect to symmetrical pints studied in [15]. Also, by putting in (11), we obtain the family defined by and further, taking , , and in , we get the set intoduced by Das and Singh [16]. For more work, see [1721].

We will assume throughout our discussion, unless otherwise stated, that

2. A Set of Lemmas

Lemma 2 (see [19]). If belongs to the class , then

Lemma 3 (see [19]). Let be analytic and starlike functions in with . Then, for , implies

3. The Main Results and Their Consequences

Theorem 4. If , then the odd functionsatisfies

Proof. If , then there exists such that This gives Because and is univalent, then by (3) we have It follows (19).

Theorem 5. A function is in the class , if and only if there exists such that

Proof. If , then it is equivalent tofor some belonging to the class . From Theorem 4, we also haveEquation (25) gives and using this in (24) provides (23). It is easy to verify that if belongs to the class and satisfies (23), then .

Theorem 6. Let . ThenAnd, for ,and

Proof. If , then by Definition 1 we havePut From (30), we can writeAlso, from (32) and (10) we have Equating the coefficients of like powers of , we haveUsing Lemma 2 and (34), we easily get which gives (27). Using Lemma 2, (27), and (35) we getIt follows that (28) and (29) hold for . We now prove (28) and (29) by using induction. Equation (36) in conjunction with Lemma 2 yieldsWe assume that (28) and (29) hold for . Then from (40) we obtainwhere we assumed In order to complete the proof, it is sufficient to show thatExpression (43) is valid for .
Let us suppose that (43) is true for . Then from (41) Thus (43) holds and hence (41) with (43) implies (28). Similarly we can prove the coefficient estimates given in (29).

Theorem 7. If , then if and only if If , then if and only if

Proof. For , we let The values of the function lie in (see (5)) since Similarly, In the case , is a disc with center and radius while it is a half plane for . Therefore for the case , , the inclusion relation holds when and This is equivalent to Furthermore, we have The domain represents an open circular disk or a half plane on the right site of the point . Therefore, and hence the result follows.

Theorem 8. If , then , where

Proof. From (58), we can easily write Let and be the numerator and denominator functions, respectively. Now we show that is starlike. Since , therefore Let Then (61) becomesSince , it follows that and Now, from (63), it follows that is starlike functions. Furthermore, Thus This implies that By Lemma 3, we have and this gives that .

Theorem 9. If in , then belongs to the class in , where

Proof. Since , it follows thatEquivalently, we havewhere is analytic in with and .
After simple computation we obtain Using (71) and (72), we can show that Thus ifNow and where we have used the well-known estimate Therefore (74) is true ifwhere Since this implies that Therefore, we have and from (78) it follows that if or Thus (74) is true if the last inequality holds.

Theorem 10. Let . Then for , ,

Proof. Let us put . Then we obtainSince is odd starlike, it follows thatFurthermore, for , one can easily obtain that Applying the last inequalities along with (87) in (86) we easily obtain (85).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors agree with the contents of the manuscript and there are no conflicts of interest among the authors.


This work is supported by National Natural Science Foundation of China (Grant no. 11571299) and Natural Science Foundation of Jiangsu Province of China (Grant no. BK20151304).