Journal of Function Spaces
Volume 2019 (2019), Article ID 6157394, 8 pages
https://doi.org/10.1155/2019/6157394

Research Article

A New Class of Analytic Functions Associated with Sălăgean Operator

Muhammad Arif,¹ Khurshid Ahmad,¹ Jin-Lin Liu,² and Janusz Sokół³

¹Department of Mathematics, Abdul Wali Khan University Mardan, 23200 Mardan, Pakistan
²Department of Mathematics, Yangzhou University, Yangzhou 225002, China
³University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland

Correspondence should be addressed to Jin-Lin Liu; jlliu@yzu.edu.cn

Received 29 June 2018; Accepted 15 January 2019; Published 3 February 2019

Academic Editor: Mitsuru Sugimoto

Copyright © 2019 Muhammad Arif et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 form

which 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 by

The 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 that

Using 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 if

where , , 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 [17–21].

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 function

satisfies

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 to

for some belonging to the class . From Theorem 4, we also have

Equation (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 . Then

And, for ,

and

Proof. If , then by Definition 1 we have

Put

From (30), we can write

Also, from (32) and (10) we have

Equating the coefficients of like powers of , we have

Using Lemma 2 and (34), we easily get

which gives (27). Using Lemma 2, (27), and (35) we get

It follows that (28) and (29) hold for . We now prove (28) and (29) by using induction. Equation (36) in conjunction with Lemma 2 yields

We assume that (28) and (29) hold for . Then from (40) we obtain

where we assumed

In order to complete the proof, it is sufficient to show that

Expression (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) becomes

Since , 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 that

Equivalently, we have

where is analytic in with and .
After simple computation we obtain

Using (71) and (72), we can show that

Thus if

Now

and

where we have used the well-known estimate

Therefore (74) is true if

where

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 obtain

Since is odd starlike, it follows that

Furthermore, 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.

Acknowledgments

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).

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