Research Article | Open Access

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

**Academic Editor:**John R. Akeroyd

#### 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.

#### References

- A. W. Goodman,
*Univalent Functions*, vol. I, II, Polygonal Publishing House, Washington, DC, USA, 1983. - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. View at: Publisher Site | Google Scholar | MathSciNet - S. Kanas and A. Wisniowska, “Conic regions and
*k*-uniform convexity,”*Journal of Computational and Applied Mathematics*, vol. 105, pp. 327–336, 1999. View at: Publisher Site | Google Scholar | MathSciNet - S. Kanas and A. Wisniowska, “Conic domains and starlike functions,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 45, no. 4, pp. 647–657, 2000. View at: Google Scholar | MathSciNet - S. Kanas, “Techniques of the differential subordination for domains bounded by conic sections,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 38, pp. 2389–2400, 2003. View at: Publisher Site | Google Scholar | MathSciNet - M. Arif, S. Mahmood, J. Sokol, and J. Dziok, “New subclass of analytic functions in conical domain associated with a linear operator,”
*Acta Mathematica Scientia, Series B: English Edition*, vol. 36, no. 3, pp. 704–716, 2016. View at: Publisher Site | Google Scholar | MathSciNet - S. Mahmood and J. Sokół, “New subclass of analytic functions in conical domain associated with ruscheweyh
*q*-differential operator,”*Results in Mathematics*, 2016. View at: Publisher Site | Google Scholar - W. Ul-Haq and S. Mahmood, “Certain properties of a class of close-to-convex functions related to conic domains,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 847287, 6 pages, 2013. View at: Publisher Site | Google Scholar - B. Pinchuk, “Functions of bounded boundary rotation,”
*Israel Journal of Mathematics*, vol. 10, pp. 6–16, 1971. View at: Publisher Site | Google Scholar | MathSciNet - K. I. Noor and S. N. Malik, “On coefficient inequalities of functions associated with conic domains,”
*Computers & Mathematics with Applications*, vol. 62, pp. 2209–2217, 2011. View at: Publisher Site | Google Scholar | MathSciNet - S. Shams, S. R. Kulkarni, and J. M. Jahangiri, “Classes of uniformly starlike and convex functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 55, pp. 2959–2961, 2004. View at: Publisher Site | Google Scholar | MathSciNet - W. Janowski, “Some extremal problems for certain families of analytic functions—I,”
*Annales Polonici Mathematici*, vol. 28, pp. 297–326, 1973. View at: Google Scholar | MathSciNet - W. Rogosinski, “On the coefficients of subordinate functions,”
*Proceedings of the London Mathematical Society*, vol. 48, pp. 48–82, 1943. View at: Publisher Site | Google Scholar | MathSciNet - S. S. Miller, “Differential inequalities and Carathéodory functions,”
*Bulletin of the American Mathematical Society*, vol. 81, pp. 79–81, 1975. View at: Publisher Site | Google Scholar | MathSciNet - S. D. Bernardi, “Convex and starlike univalent functions,”
*Transactions of the American Mathematical Society*, vol. 135, pp. 429–446, 1969. View at: Publisher Site | Google Scholar | MathSciNet - R. J. Libera, “Some classes of regular univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 16, pp. 755–758, 1965. View at: Publisher Site | Google Scholar | MathSciNet - A. E. Livingston, “On the radius of univalence of certain analytic functions,”
*Proceedings of the American Mathematical Society*, vol. 17, pp. 352–357, 1966. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

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.