q -Generalized Linear Operator on Bounded Functions of Complex Order

: This article presents a q -generalized linear operator in Geometric Function Theory (GFT) and investigates its application to classes of analytic bounded functions of complex order S q ( c ; M ) and C q ( c ; M ) where 0 < q < 1, 0 (cid:54) = c ∈ C , and M > 12 . Integral inclusion of the classes related to the q -Bernardi operator is also proven.


Introduction
Quantum calculus or q-calculus is attributed to the great mathematicians L.Euler and C. Jacobi, but it became popular when Albert Einstein used it in quantum mechanics in his paper [1] published in 1905. F.H. Jackson [2,3] introduced and studied the q-derivative and q-integral in a proper way. Later, quantum groups gave the geometrical aspects to q-calculus. It is pertinent to mention that q-calculus can be considered an extension of classical calculus discovered by I. Newton and G.W. Leibniz. In fact, the operators defined as: and: where z ∈ C and h > 0 are the h-derivative and q-derivative, respectively, where h is Planck's constant, are related as: q = e ih = e 2πih where h = h/2π. Srivastava [4] applied the concepts of q-calculus by using the basic (or q-) hypergeometric functions in Geometric Function Theory (GFT). Ismail [5] and Agarwal [6] introduced the class of q-starlike functions by using the q-derivative. The q-close-to-convex functions were defined in [7], and Sahoo and Sharma [8] obtained several interesting results for q-close-to-convex functions. Several convolution and fractional calculus q-operators were defined by the researchers, which were reposited by Srivastava in [9]. Darus [10] defined a new differential operator called the q-generalized operator by using q-hypergeometric functions. Let A be the class of functions of the form: analytic in the open unit disc E = {z : |z| < 1}. Let f (z) be given by (1) and g(z) defined as: The Hadamard product (or convolution) of f and g is defined by: A sequence {b k } ∞ k=1 of complex numbers is a subordinating factor if, whenever f (z) = ∑ ∞ k=1 a k z k , a 1 = 1 is regular, univalent, and convex in E, we have ∑ ∞ n=1 b n a n z n ≺ f (z), z ∈ E [11]. We recall some basic concepts from q-calculus that are used in our discussion and refer to [2,3,12] for more details.
A subset B ⊂ C is called q-geometric if zq ∈ B whenever z ∈ B, and it contains all the geometric sequences zq k ∞ 0 . In GFT, the q-derivative of f (z) is defined as: and d q f (0) = f (0). For a function g(z) = z k , the q-derivative is: where [k] = 1−q k 1−q = 1 + q + q 2 + .... + q k−1 . We note that as q → 1 − , d q f (z) → f (z), which is the ordinary derivative. From (1), we deduce that: Let f (z) and g(z) be defined on a q-geometric set B. Then, for complex numbers a, b, we have: Jackson [2] introduced the q-integral of a function f , given by: provided that the series converges. For any non-negative integer n, the q-number shift factorial is defined as: Let λ ∈ R and n ∈ N; the q-generalized Pochhammer symbol is defined as: The q-Gamma function is defined for λ > 0 as: with ( n 2 ) = n(n−1) 2 and l, m ∈ N 0 = N ∪ {0}. Here, the q-shifted factorial is defined for a ∈ C as: , and by using the property (q a ) k = Γ q (a + k) (1 − q) k /Γ q (a), from (2), we get the function, In [13], the q-Srivastava-Attiya convolution operator is defined as: (a ∈ C \ Z − 0 ; s ∈ C when |z| < 1; Re(s) > 1 when |z| = 1). Using convolution, the operator D s q,a,λ for λ > −1 is defined as: where: It is a convergent series with a radius of convergence of one. We observe that D 0 q,a,0 f (z) = f (z) and D 1 q,0,0 f (z) = zd q f (z). The operator D s q,a,λ reduces to known linear operators for different values of parameters a, s, and λ as: (i) If q → 1 − , it reduces to the operator D s a,λ discussed by Noor et al. in [14]. (ii) For s = 0, it is a q-Ruscheweyh differential operator [15]. (iii) If s = −1, λ = 0, and q → 1 − , it is an Owa-Srivastava integral operator [16]. (iv) If s ∈ N 0 , a = 1, λ = 0, and q → 1 − , it reduces to the generalized Srivastava-Attiya integral operator [17]. (v) If s ∈ N 0 , a = 0, λ = 0, it is a q-Salagean differential operator [18]. (vi) For s, λ ∈ N 0 , and a = 0, it is the operator defined in [19].
The following identities hold for the operator D s q,a,λ f (z), Let P(q) be the class of functions of the form p(z) = 1 + c 1 z + c 2 z 2 + ...., analytic in E, and satisfying: It is known from [20] that p ∈ P(q) implies p(z) ≺ 1+z 1−qz . It follows immediately that Re p(z) > 0, The classes of bounded q-starlike functions S q (c, M) and bounded q-convex functions C q (c, M) of complex order c were defined in [21], respectively, as: or equivalently, The class of bounded q-convex functions C q (c, M) of complex order c is defined as: or equivalently, Using the operator D s q,a,λ f (z), we now define the following new classes S q,a,s,λ (c, M) and C q,a,s,λ (c, M) as: Special cases: (i) If c = 1, m = 1, and q → 1 − , then S q,a,s,λ (c, M) reduces to class S s (a, λ) discussed in [22].
reduces to class ST q studied by Noor [24].
, and q → 1 − , then S q,a,s,λ (c, M) becomes special cases of Janowski β-spiral like functions of complex order S β (A, B, a) discussed in [25].
A function f ∈ A is in the class S q,a,s,λ (c, M) if and only if: where A = c(1 + m) − m and B = −m. The class C q,a,s,λ (c, M) is defined as: It is easy to see that f ∈ C q,a,s,λ (c, M) ⇔ zd q f ∈ S q,a,s,λ (c, M). In order to develop results for the classes S q,a,s,λ (c, M) and C q,a,s,λ (c, M), we need the following: . Let β and γ be complex numbers with β = 0, and let h(z) be regular in E with h(0) = 1 and

Lemma 2 ([11]
). The sequence {b n } ∞ n=1 is a subordinating factor sequence if and only if: S q,a,s,λ (c, M) and C q,a,s,λ (c, M)

Properties of Classes
We start the section with the necessary and sufficient condition for a function to be in the class S q,a,s,λ (c, M). Theorem 1. Let f ∈ A. Then, f ∈ S q,a,s,λ (c, M) if and only if: Proof. Let us assume first that Inequality (6) holds. To show f ∈ S q,a,s,λ (c, M), we need to prove Inequality (5). [k+a] s .
Theorem 3. Let f i with i = 1, 2, ..., ν belong to the class S q,a,s,λ (c, M). The arithmetic mean h of f i is given by: (8) belonging to class S q,a,s,λ (c, M).
Proof. From (8), we can write: Since f i ∈ S q,a,s,λ (c, M) for every i = 1, 2, ..., v, using (6) and (9), we have: and this completes the proof. Now, we give the subordination relation for the functions in class S q,a,s,λ (c, M) by using the subordination theorem.
Let us consider the function: which is a member of S q,a,s,λ (c, M). Then. by using (10), we have: It is easily verified that:
Now, we discuss the inclusion results pertaining to classes S q,a,s,λ (c, M) and C q,a,s,λ (c, M) in reference to parameters s and λ.

Theorem 6.
For any complex number s, C q,a,s+1,λ (c, Proof. It is obvious from the fact f ∈ C q,a,s,λ (c, M) ⇔ zd q f ∈ S q,a,s,λ (c, M).
In [30], the q-Bernardi integral operator L b f (z) is defined as: a k z k , b = 1, 2, 3, .... Now, we apply the generalized operator D s q,a,λ on L b f (z) as: The identity relation of D s q,a,λ (L b f (z)) is given as: The following theorems are the integral inclusions of the classes S q,a,s,λ (c, M) and C q,a,s,λ (c, M) with respect to the q-Bernardi integral operator.
We will show: which would prove L b g(z) ∈ S q,a,s,λ (c, M). From the identity relation (19), after some calculations, we have: zd q (D s q,a,λ L b g(z)) D s q,a,λ L b g(z) D s q,a,λ g(z) (D s q,a,λ L b g(z)) After some calculations, we have: D s q,a,λ g(z) D s q,a,λ L b g(z) q,a,λ L b g(z)) D s q,a,λ L b g(z)