ON NEW SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH MODIFIED Q-OPERATOR

In this article, we introduce new subclasses of harmonic univalent functions associated with the q-difference operator. The modified q-Srivastava-Attiya operator is defined and certain applications of this operator are discussed. We investigate the sufficient condition, distortion result, extreme points and invariance of convex combination of the elements of the subclasses.


Introduction
A real-valued function u (x, y) is said to be harmonic in a domain D if it has continuous second order partial derivatives in D and satisfies u xx + u yy = 0.
We say that a continuous f : Ω (⊂ C) → C defined by f (z) = u (x, y) +iv (x, y) is harmonic if both u (x, y) and v (x, y) are real harmonic in Ω. It is observed that every harmonic function f in any simply connected domain Ω can be written as f (z) = h(z) + g(z), where h and g are analytic in Ω, and are called, respectively, the analytic and co-analytic parts of f .
It is clear that, when g(z) is identically zero, the class H reduces to the class A of normalized analytic functions in U.
Due to Lewy [14], A function f ∈ H is locally univalent and sense-preserving in U if and only if |h (z)| > |g (z)| , for z ∈ U.
We denote by S H the subclass of H consisting of all sense-preserving univalent harmonic functions f .
Firstly, Clunie et al. [5] was discussed some geometric properties of the class S H and its subclasses. Later on, several authors contributed in the study of subclasses of the class S H , for example, see [1,3,6,7,9,11,19,20].
The theory of q-calculus operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as geometric function theory of complex analysis. The fractional q-calculus is the q-extension of the ordinary fractional calculus and dates back to early 20 th century (e.g. see [10] or [2]). For 0 < q < 1, the q-difference operator was introduced by Jackson [10] and is defined by for q ∈ (0, 1) and h ∈ A with h(z) = z + ∞ n=2 a n z n . It is clear that lim is the ordinary derivative of the function. It can easily be seen that for n ∈ N = {1, 2, 3, ..} and z ∈ U Recently, in [18], Shah and Noor introduced the q-analogue of Srivastava-Attiya operator J s q,b : A → A by where h ∈ A, s ∈ C and b ∈ C\Z − 0 . It is noted that for q → 1 − in (1.3), then the integral operator studied by the authors in [21] is deduced. Moreover, for particular choices of s and b, the operator J s q,b reduces to the q-Alexander, q-Libera and q-Bernardi operators defined in [17]. Jahangiri [12] was the first who introduced the q-analogue of complex harmonic functions and studied various geometric properties. Nowadays, certain subclasses of S H associated with operators and q-operators were discussed by the prominent researchers, like [8,12,13,15,16,22]. In motivation of above said literature, first we modify the q-Srivastava-Attiya operator and then we define some new subclasses of S H .
For f = h + g given by (1.1), we define where J s q,b h(z) is given by (1.3) and It is observed that, if co-analytic part of f = h + g is identically zero, then the modified q-Srivastava-Attiya operator defined by (1.4) turn out to be the q-Srivastava-Attiya operator introduced in [18].
Definition 1.2. Let f = h + g ∈ S H , and is given by Also, we define HS where S H denote the subclass of S H consisting of functions given by (1.5).

Main Results
Theorem 2.1. Let a function f = h + g ∈ S H given by (1.1) and satisfies . This result is sharp.
Proof. We wish to show that f = h + g ∈ S H satisfies (1.6), whenever the coefficients of f satisfies (2.1).
We use the fact that (w) ≥ ξ if and only if |1 − ξ + w| ≥ |1 + ξ − w|. So it suffices to show that or equivalently, Using (1.3) and (1.4), from the left hand side, we get [n] q a n z n−1 + ∞ n=1 The above expression is nonnegative by (2.1). Hence f ∈ HS s,b q (α, β). The coefficient bound, given by (2.1), is sharp for the harmonic function For different choices of parameters, we deduce certain new results as following.
If we take γ = 1 in Corollary 2.2 then we obtain.
Proof. The sufficient condition is obvious from the Theorem 2.1, because HS s,b q (γ, λ, β) ⊂ HS s,b q (γ, λ, β). We need to prove the necessary condition only, that is, if f ∈ HS s,b q (γ, λ, β), then the coefficients of the function f = h + g satisfy the inequality 2.2.
Let f ∈ HS s,b q (γ, λ, β). Then, by the definition of HST s,b q (γ, λ, β), we have Equivalently, we can write (2.3) as Substituting f = h + g in (2.4) and employing (1.4) along with (1.5), and also some computation yields This implies The above required condition must hold for all values of z in U. Upon choosing the values of z on the positive real axis where 0 ≤ z = r < 1, we must have If the inequality (2.2) does not hold, then the numerator in (2.5) is negative for r sufficiently close to 1.
Hence there exists z 0 = r 0 in (0, 1) for which the quotient in (2.5) is negative. This contradicts the required condition for f ∈ HST s,b q (γ, λ, β) and so the proof is complete.
Next, we want to discuss the distortion bounds for the function f ∈ HS s,b q (γ, λ, β), which yields a covering result for this class. q (γ, λ, β) and |z| = r < 1, then Proof. Let f ∈ HS s,b q (γ, λ, β). Taking absolute value of f , we get where T is given by (2.6). Hence this is the required right hand inequality. Similarly, one can easily prove the required left hand inequality.
By making use of the left hand inequality of the above theorem and letting r → 1, we obtain In particular, we obtain the covering results for the subclasses of harmonic functions defined in Definition 1.1 and its special cases. Now, our task is to determine the extreme points of closed convex hulls of HS  Proof. We assume function f as given by (2.7) Equating (2.8) with (1.5), we get |a n | = X n R n and |b n | = Y n R n .

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.