Journal of Function Spaces
Volume 2017 (2017), Article ID 6095293, 7 pages
https://doi.org/10.1155/2017/6095293

Research Article

A New Subclass of -Janowski Type Functions Associated with Ruscheweyh Derivative

Shahid Mahmood,¹ Sarfraz Nawaz Malik,² Saima Mustafa,³ and S. M. Jawwad Riaz²

¹Department of Mechanical Engineering, Sarhad University of Science & IT, Landi Akhun Ahmad, Hayatabad Link, Ring Road, Peshawar, Pakistan
²Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan
³Department of Statistics & Mathematics, PMAS-Arid Agriculture University, Rawalpindi, Pakistan

Correspondence should be addressed to Shahid Mahmood; shahidmahmood757@gmail.com

Received 25 February 2017; Accepted 13 April 2017; Published 3 May 2017

Academic Editor: John R. Akeroyd

Copyright © 2017 Shahid Mahmood 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.

We introduce and investigate a new subclass of analytic functions using Ruscheweyh derivative. We derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the general class, which we have introduced and studied in this article. We also observe that this class is preserved under the Bernardi integral transform.

1. Introduction

Let denote the class of functions of the form

which are analytic in the unit disc Also let and denote the well-known classes of starlike and convex functions of order , respectively. For details, see [1]. For any two analytic functions and with

their convolution (Hadamard product) is given by

In 1975, using the concept of convolution, Ruscheweyh [2] introduced a linear operator defined by

with

where is a Pochhammer symbol given as

It is obvious that and

The following identity can easily be established:

The operator is called the Ruscheweyh derivative of ; see [2].

Suppose also that, for , the classes and denote the well-known classes of -uniformly convex and -starlike functions, respectively. These classes were introduced by Kanas and Wisniowska [3, 4]. For some details see [3–5].

Consider the domain

For fixed , represents the conic region bounded successively by the imaginary axis , the right branch of a hyperbola , a parabola , and an ellipse . This domain was studied by Kanas [3–5]. The function , with , plays the role of extremal and is given by

where , and is chosen such that , with being Legendre’s complete elliptic integral of the first kind and being complementary integral of (see [3, 4, 6–8]). Let denote the class of all those functions which are analytic in with and for . Clearly it can be seen that where is the class of functions with positive real part (see [1, 9]). More precisely

and, for , we have

where

So we can write

Definition 1 (see [10]). A function is said to be in the class , if and only if

where is defined by (10) and Geometrically, the function takes all values from the domain which is defined as

or equivalently

The domain retains the conic domain inside the circular region defined by The impact of on the conic domain changes the original shape of the conic regions. The ends of hyperbola and parabola get closer to each other but never meet anywhere and the ellipse gets the shape of oval. When , the radius of the circular disk defined by tends to infinity, consequently the arms of hyperbola and parabola expand, and the oval turns into ellipse.
With the help of the above Ruscheweyh derivative, we now define the following class.

Definition 2. A function of the form (1) is in the class if and only if

or equivalently,

with being given by (14), , , , and
For different permissible choices of parameters, we obtain several known as well as new subclasses of the class of analytic functions as special cases; for example,

(i)For and , we obtain , and for , we have the class These classes are recently introduced and studied by [10].(ii), the well-known classes of -uniformly convex and -starlike functions, respectively, introduced by Kanas and Wisniowska [3, 4].(iii), the classes, introduced by Shams et al. in [11].(iv), the well-known classes of Janowski starlike and Janowski convex functions, respectively, introduced by Janowski [12].

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

2. Preliminary Results

We need the following lemmas to obtain our results.

Lemma 3 (see [5]). Let with any complex numbers with and If is analytic in with and satisfies

and is an analytic solution of

then is univalent,

and is the best dominant of (19), where is given as

Lemma 4 (see [10]). Let Then

where

Lemma 5 (see [13]). Let in . If is univalent in and is convex, then

Lemma 6 (see [14]). Let , and , where

Let in with and let

Then

and the bound in (29) is sharp, the extremal functions being

with

3. Main Results

Theorem 7. A function and of the form (1) is in the class , if it satisfies the condition

Proof. Let we note that

because from (32) it follows that

To show that it suffices that

From (33), we have

The last expression is bounded above by if

and this completes the proof.

When we put and in the above theorem, we obtain the following known result, proved by Noor and Malik in [10].

Corollary 8. A function and of the form (1) is in the class , if it satisfies the condition

For , with , , and , Theorem 7 reduces to the following known result, proved by Shams et al. [11].

Corollary 9. A function and of the form (1) is in the class , if it satisfies the condition

Theorem 10. Let Then

where

where and are given by (24) and (5).

Proof. Set

so that Let Then (42) can be written as

which implies that

Using the coefficient estimates for the class (see [10]), we obtain

For ,

Therefore (40) holds for Assume that (40) is true for and consider

Therefore, the result is true for . Using mathematical induction, (40) holds true for all

Corollary 11 (see [10]). If , then

where is defined by (8).

When , , , and , we obtain the following coefficient inequality for the class , introduced by Kanas and Wisniowska [4].

Corollary 12. If , then

This result is sharp.

By taking the values with , , and , we obtain the coefficient inequality of the class , introduced by Shams et al. [11].

Corollary 13. If , then

This result is sharp.

Theorem 14. For real , let Then for

Proof. Suppose and set

where is analytic in with Then simple computations, together with (51) and (8), yield

with Since , it follows that

or, equivalently,

Since and is a convex set (see [10]), it follows that , with , belong to in and hence We now use the Lemma 3 with , and to obtain and hence This complete the proof.

For a function , we consider the integral operator

The operator was introduced by Bernardi [15] for . In particular, the operator was studied earlier by Libera [16] and Livingston [17].

Theorem 15. Let and let be defined by (55). Then

Proof. Let

where is analytic in , Then using (8), we have

where Simple computation and use of (8), (55), and (56), we have

Apply Lemma 3 with and to obtain and consequently

Corollary 16. Let , , and and let be defined by (55). Then , where , and

Proof. Proceeding as in Theorem 15, it follows from (58) that

Applying Lemma 6, we obtain , where is given by (59). This proves that in

Theorem 17. If is of the form (1) belonging to and, where is the integral operator defined by (55), then

Proof. From (55), we obtain

Using the series for the functions and , we obtain

and thus

From the above we have

Using the estimates from Theorem 10, we obtain the required result.

Conflicts of Interest

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

Acknowledgments

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

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