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

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 formwhich 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 withwhere 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 bywhere , 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 ifwhere is defined by (10) and Geometrically, the function takes all values from the domain which is defined as or equivalentlyThe 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 satisfiesand 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 Thenand 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 thatbecause 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 Thenwherewhere and are given by (24) and (5).

Proof. Setso 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 setwhere is analytic in with Then simple computations, together with (51) and (8), yieldwith 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. Letwhere is analytic in , Then using (8), we have where Simple computation and use of (8), (55), and (56), we haveApply 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.