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 byIt can easily be seen that for and whereFor any nonnegative integer , the -number shift factorial is defined byAlso the -generalized Pochhammer symbol for is given by
For , we define the function by
where the function is given byIt 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 bywithFrom (10), we can easily get the identityWe note that , , andThis 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 satisfieswhere the notion “” denotes the familiar subordinations.
Equivalently, a function is in the class , if and only ifWe will assume throughout our discussion, unless otherwise stated, thatand 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, considerwhere 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 haveNow 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 Thenwith and
Proof. Let and settingwithequivalently, we can writeor in other way, we haveThus we can rewriteand further by simple computation of integration, the proof is completed.
Theorem 5. Let and have the formThen , where
Proof. By the virtue of Theorem 3, one can writeThereforehoweverthen 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 writeTo prove that , we need to show thatFor this, considerwhere we have used inequality (18). Hence the result follows.
Theorem 7. Let with belong to the class Then the arithmetic mean of is given byand is also in the class
Proof. From (37), we can writeSince for every , using (38) and (18), we haveand 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 showUsing (1) along with some simple computation yieldsSince , from (18), we can easily obtainNow inequality (42) will be true, if the following holds:which implies thatand thus we get the needed result.
Theorem 9. Let and in , and this satisfiesThen
Proof. Since in , therefore let us define the function byBy the virtue of identity (12), we obtainTherefore, using (46), we haveand 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).