Journal of Function Spaces
Volume 2018 (2018), Article ID 3818915, 5 pages
https://doi.org/10.1155/2018/3818915

Research Article

A Subfamily of Univalent Functions Associated with -Analogue of Noor Integral Operator

Muhammad Arif,¹ Miraj Ul Haq,¹ and Jin-Lin Liu²

¹Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan
²Department of Mathematics, Yangzhou University, Yangzhou 225002, China

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

Received 14 September 2017; Accepted 8 January 2018; Published 1 February 2018

Academic Editor: John R. Akeroyd

Copyright © 2018 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 objective of the present paper is to define a new subfamily of analytic functions using subordinations along with the newly defined -Noor integral operator. We investigate a number of useful properties such as coefficient estimates, integral representation, linear combination, weighted and arithmetic means, and radius of starlikeness for this class.

1. Introduction and Definitions

In recent years, -analysis (-calculus) has motivated the researchers a lot due to its numerous applications in mathematics and physics. Jackson [1, 2] was the first to give some application of -calculus and also introduced the -analogue of derivative and integral operator. Later on, Aral and Gupta [3, 4] defined the -Baskakov-Durrmeyer operator by using -beta function while in papers [5, 6] the authors discussed the -generalization of complex operators known as -Picard and -Gauss-Weierstrass singular integral operators. Using convolution of normalized analytic functions, Kanas and Raducanu [7] defined -analogue of Ruscheweyh differential operator and studied some of its properties. The application of this differential operator was further studied by Aldweby and Darus [8] and Mahmood and Sokół [9]. The aim of the current paper is to define a -analogue of the Noor integral operator involving convolution concepts and then give some interesting applications of this operator.

Let us denote the open unit disk by and the symbol denotes the family of those analytic functions which has the following Taylor series representation:

For two functions and that are analytic in and have the form (1), we define the convolution of these functions by

For , the -derivative of a function is defined by

It can easily be seen that for and

where

For any nonnegative integer , the -number shift factorial is defined by

Also the -generalized Pochhammer symbol for is given by

For , we define the function by

where the function is given by

It is quite clear that the series defined in (9) is convergent absolutely in . Using the definition of -derivative along with the idea of convolutions, we now define the integral operator by

with

From (10), we can easily get the identity

We note that , , and

This shows that, by taking , the operator defined in (10) reduces to the familiar Noor integral operator introduced in [10, 11]. Also for more details on the -analogue of differential and integral operators, see the work [12–14].

Motivated from the work studied in [7, 15–17], we now define subfamilies of the set by using the operator as follows.

Definition 1. Let and Then the function is in the class if it satisfies

where the notion “” denotes the familiar subordinations.
Equivalently, a function is in the class , if and only if

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

and all coefficients are positive.

We need the following result in the proof of a result.

Lemma 2 (see [18]). Let Then

2. Main Results

Theorem 3. Let be given by (1). Then the function is in the family , if and only if

Proof. Let us assume first that inequality (18) holds. To show , we only need to prove the inequality (15). For this, consider

where we have used (4), (10), and (18) and this completes the direct part. Conversely, let be of the form (1). Then from (15) along with (10), we have, for ,

Since , we have

Now we choose values of on the real axis such that is real. Upon clearing the denominator in (21) and letting through real values, we obtain the required inequality (18).

Theorem 4. Let Then

with and

Proof. Let and setting

with

equivalently, we can write

or in other way, we have

Thus we can rewrite

and further by simple computation of integration, the proof is completed.

Theorem 5. Let and have the form

Then , where

Proof. By the virtue of Theorem 3, one can write

Therefore

however

then Hence the proof is complete.

Theorem 6. If and belong to , then their weighted mean is also in , where is defined by

Proof. From (33), we can easily write

To prove that , we need to show that

For this, consider

where we have used inequality (18). Hence the result follows.

Theorem 7. Let with belong to the class Then the arithmetic mean of is given by

and is also in the class

Proof. From (37), we can write

Since for every , using (38) and (18), we have

and this completes the proof.

Theorem 8. Let Then is in the family of starlike functions of order for , where

Proof. Let . To prove , we only need to show

Using (1) along with some simple computation yields

Since , from (18), we can easily obtain

Now inequality (42) will be true, if the following holds:

which implies that

and thus we get the needed result.

Theorem 9. Let and in , and this satisfies

Then

Proof. Since in , therefore let us define the function by

By the virtue of identity (12), we obtain

Therefore, using (46), we have

and now, using Lemma 2, we have

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 (Grant no. BK20151304).

References

F. H. Jackson, “On q-functions and a certain difference operator,” Earth and Environmental Science Transactions of the Royal Society of Edinburgh, vol. 46, no. 2, pp. 253–281, 1909.

F. H. Jackson, “On q-definite integrals,” The Quarterly J. Pure Appl. Math, vol. 41, pp. 193–203, 1910.

A. Aral and V. Gupta, “Generalized q-Baskakov operators,” Mathematica Slovaca, vol. 61, no. 4, pp. 619–634, 2011.

A. Aral and V. Gupta, “On q-Baskakov type operators,” Demonstratio Mathematica, vol. 42, no. 1, pp. 109–122, 2009.

G. A. Anastassiou and S. G. Gal, “Geometric and approximation properties of generalized singular integrals in the unit disk,” Journal of the Korean Mathematical Society, vol. 23, no. 2, pp. 425–443, 2006.

A. Aral, “On the generalized Picard and Gauss Weierstrass singular integrals,” Journal of Computational Analysis and Applications, vol. 8, no. 3, pp. 249–261, 2006.

S. a. Kanas and D. Raducanu, “Some class of analytic functions related to conic domains,” Mathematica Slovaca, vol. 64, no. 5, pp. 1183–1196, 2014.

H. Aldweby and M. Darus, “Some subordination results on q-analogue of Ruscheweyh differential operator,” Abstract and Applied Analysis, vol. 2014, Article ID 958563, 2014.

S. Mahmood and J. Sokół, “New subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator,” Results in Mathematics, vol. 71, pp. 1–13, 2017.

K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, no. 1-2, pp. 71–80, 1999.

K. I. Noor and M. A. Noor, “On integral operators,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 341–352, 1999.

I. Aldawish and M. Darus, “Starlikeness of q-differential operator involving quantum calculus,” Korean Journal of Mathematics, vol. 22, no. 4, pp. 699–709, 2014.

H. Aldweby and M. Darus, “A subclass of harmonic univalent functions associated with q-analogue of Dziok-Srivastava operator,” ISRN Mathematical Analysis, vol. 2013, Article ID 382312, 6 pages, 2013.

A. Mohammed and M. Darus, “A generalized operator involving the q-hypergeometric function,” Matematicki Vesnik, vol. 65, no. 4, pp. 454–465, 2013.

S. Agrawal and S. K. Sahoo, “A generalization of starlike functions of order alpha,” Hokkaido Mathematical Journal, vol. 46, no. 1, pp. 15–27, 2017.

H. Aldweby and M. Darus, “Integral operator defined by q-analogue of Liu-Srivastava operator,” Studia. Universitatis Babe-Bolyai Mathematica, vol. 58, no. 4, pp. 529–537, 2013.

T. M. Seoudy and M. K. Aouf, “Coefficient estimates of new classes of q-starlike and q-convex functions of complex order,” Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 135–145, 2016.

M. S. Liu, “On a subclass of p-valent close to convex functions of type α and order β,” Journal of Mathematical Study, vol. 30, no. 1, pp. 102–104.