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

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.


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 D = { ∈ C : | | < 1} and the symbol A denotes the family of those analytic functions which has the following Taylor series representation: For two functions and that are analytic in D and have the form (1), we define the convolution of these functions by For 0 < < 1, the -derivative of a function ∈ A is defined by It can easily be seen that for ∈ N fl {1, 2, 3, . . .} and ∈ D where For any nonnegative integer , the -number shift factorial is defined by For > −1, we define the function F −1 , +1 ( ) by where the function F , +1 ( ) is given by It is quite clear that the series defined in (9) is convergent absolutely in D. Using the definition of -derivative along with the idea of convolutions, we now define the integral operator I : A → A by From (10), we can easily get the identity We note that I 0 ( ) = ( ), I 1 ( ) = ( ), and This shows that, by taking → 1 − , 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][13][14]. Motivated from the work studied in [7,[15][16][17], we now define subfamilies of the set A by using the operator I as follows.
where the notion "≺" denotes the familiar subordinations.
Equivalently, a function ∈ A is in the class Q ( , , ), 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.

Main Results
Theorem 3. Let ∈ A be given by (1). Then the function is in the family Q ( , , ), if and only if Proof. Let us assume first that inequality (18) holds. To show ∈ Q ( , , ), we only need to prove the inequality (15).
where we have used (4), (10), and (18) and this completes the direct part. Conversely, let ∈ Q ( , , ) be of the form (1). Then from (15) along with (10), we have, for ∈ D,  Now we choose values of on the real axis such that (I ( ))/I ( ) is real. Upon clearing the denominator in (21) and letting → 1 − through real values, we obtain the required inequality (18).
Proof. Let ∈ Q ( , , ) and setting (I ( )) with equivalently, we can write or in other way, we have Thus we can rewrite (I ( )) and further by simple computation of integration, the proof is completed.
Then ∈ Q ( , , ), where Proof. By the virtue of Theorem 3, one can write Therefore then ∈ Q ( , , ). Hence the proof is complete.

Theorem 6.
If and belong to Q ( , , ), then their weighted mean ℎ is also in Q ( , , ), where ℎ is defined by Proof. From (33), we can easily write To prove that ℎ ∈ Q ( , , ), we need to show that For this, consider where we have used inequality (18). Hence the result follows. Journal of Function Spaces Theorem 7. Let with = 1, 2, . . . , belong to the class Q ( , , ). Then the arithmetic mean ℎ of is given by and is also in the class Q ( , , ).

(40)
Proof. Let ∈ Q ( , , ). To prove ∈ S * ( ), we only need to show Using (1) along with some simple computation yields Since ∈ Q ( , , ), 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.

Conflicts of Interest
The authors agree with the contents of the manuscript and there are no conflicts of interest among the authors.