Certain subclass of analytic functions related with conic domains and associated with Salagean q-di ff erential operator

Abstract: In our present investigation, by using Salagean q-differential operator we introduce and define new subclass k − US(q, γ,m), γ ∈ C\{0}, and studied certain subclass of analytic functions in conic domains. We investigate the number of useful properties of this class such structural formula and coefficient estimates Fekete–Szego problem, we give some subordination results, and some other corollaries.


Introduction
Let A denotes the class of all function f (z) which are analytic in the open unit disk E = {z ∈ C : |z| < 1} and normalized by f (0) = 0 and f (0) = 1, so each f ∈ A has the Maclaurin's series expansion of the form: (1.1) A function f : E → C is called univalent on E if f (z 1 ) = f (z 2 ) for all z 1 = z 2 , z 1 , z 2 ∈ E. Let S ⊂ A be the class of all functions which are univalent in E (see [3]).Recall D ⊂ C is said to be a starlike with respect to the point d 0 ∈ D if and only if the line segment joining d 0 to every other point d ∈ D lies entirely in D, while the set D is said to be convex if and only if it is starlike with respect to each of its points.By S * and K we means the subclasses of S composed of starlike and convex functions.A function f ∈ A is said to be starlike of order α, 0 ≤ α < 1, if A function f ∈ A is said to be convex of order α, 0 ≤ α < 1, if In 1991, Goodman [4] introduced the class UCV of uniformly convex functions which was extensively studied by Ronning and independently by Ma and Minda [1,2].A more convenient characterization of class UCV was given by Ma and Minda as: In 1999, Kanas and Wisniowska [5,6] introduced the class k−uniformly convex functions, k ≥ 0, denoted by k − UCV and a related class k − ST as: The class k − UCV was discussed earlier in [7], see also [8] with same extra restriction and without geometrical interpretation by Bharati et.al [8].In 1985, Nasr et al., studied a natural extension of classical starlikness in order terminology.We say that a function f (z Several author investigated the properties of the class, S * k,γ and their generalizations in several directions for detail study see [4,6,9,10,11,12,13].The convolution or Hadamard product of two function f and g is denoted by f * g is defined as where f (z) is given by (1.1) and g(z) = ∞ n=2 b n z n , (z ∈ E).If f (z) and g(z) are analytic in E, we say that f (z) is subordinate to g(z), written as f (z) ≺ g(z), if there exists a Schwarz function w(z), which is analytic in E with w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)).Furthermore, if the function g(z) is univalent in E, then we have the following equivalence, see [3,14].

AIMS Mathematics
Volume 2, Issue 4, 622-634 Note that the q-difference operator plays an important role in the theory of hypergeometric series and quantum theory, number theory, statistical mechanics, etc.At the beginning of the last century studies on q-difference equations appeared in intensive works especially by Jackson [33], Carmichael [32], Mason [34], Adams [31] and Trjitzinsky [35].Research work in connection with function theory and q-theory together was first introduced by Ismail et al. [36].Till now only non-significant interest in this area was shown although it deserves more attention.Many differential and integral operators can be written in term of convolution, for details we refer [21].
It is worth mentioning that the technique of convolution helps researchers in further investigation of geometric properties of analytic functions.
For any non-negative integer n, the q-integer number n denoted by [n] q , is defined by For non-negative integer n the q-number shift factorial is defined by We note that when q → 1, [n]! reduces to classical definition of factorial.In general, for a non-integer number t, [t] q is defined by [t] q = 1−q t 1−q , [0] q = 0. Throughout in this paper, we will assume q to be a fixed number between 0 and 1 The q-difference operator related to the q-calculus was introduced by Andrews et al. (see in [30] CH 10).For f ∈ A, the q-derivative operator or q-difference operator is defined as.
Recently, Govindaraj and Sivasubramanian defined Salagean q-differential operator [28] as: Let f ∈ A, let Salagean q-differential operator Making use of (1.2) and (1.3), the power series of S m q f (z) for f of the form (1.1) is given by which is the familiar Salagean derivative [29].
Taking motivation from the work shahid et.al [23], we introduce new subclass k − US(q, γ, m), of analytic functions with the theory of q-calculus by using Salagean q-differential operator.
By taking specific values of parameters, we obtain many important subclasses studied by various authors in earlier papers.Here we inlist some of them.

Geometric Interpretation
where Since p k,γ (z) is convex univalent, so above definition can be written as where (1.6) The boundary ∂Ω k,γ of the above set becomes the imaginary axis when k = 0, while a hyperbola when 0 < k < 1.For k = 1 the boundary ∂Ω k,γ becomes a parabola and it is an ellipse when k > 1 and in this case where and t ∈ (0, 1) is chosen such that k = cosh (πK (t)/(4K(t))).Here K(t) is Legender's complete elliptic integral of first kind and ) and K (t) is the complementary integral of K (t) for details see [5,6,14,17].Moreover, p k,γ (E) is convex univalent in E, see [5,6].All of these curves have the vertex at the point k+γ k+1 .

Set of Lemmas
Each of the following lemmas will be needed in our present investigation.

Main Results
In this section, we will prove our main results.
Proof.If f (z) ∈ k − US(q, γ, m) then using the identity (1.5), we obtain For some function w(z) is analytic in E with w(0) = 0 and |w(z)| < 1. Integrating (3.3) and after some simplification we have This proves (3.1).Noting that the univalent function p k,γ (z) maps the disk |z| < ρ (0 < ρ ≤ 1) onto a region which is convex and symmetric with respect to the real axis, we see Using (3.4) and (3.5) gives for z ∈ E. Consequently, subordination (3.4) leads us to this completes the proof. and ) where p(z) is analytic in E and p(0) = 1.Let p(z) = 1 + ∞ n=1 c n z n and S m q f (z) is given by (1.4).Then (3.8) becomes Now comparing the coefficients of z n , we obtain which implies [ j] m q a j h n− j .
Using the results that |c n | ≤ |Q 1 | given in ( [17]), we have Let us take δ = |Q 1 | .Then we have For n = 2 in (3.9), we have which shows that (3.7) holds for n = 2.To prove (3.7) we use principle of mathematical induction, for this, consider the case n = 3 Using (3.10), we have which shows that (3.7) holds for n = 3.Let us assume that (3.7) is true for n ≤ t, that is, which proves the assertion of theorem n = t + 1. Hence (3.7) holds for all n, n ≥ 3.This completes the proof.
Proof.Let f (z) ∈ k−US(q, γ, m), then there exists Schwarz function w(z), with w(0) = 0 and |w(z)| < 1 such that z∂ q S m q f (z) S m q f (z) Let p(z) ∈ P be a function defined as

This gives
z∂ q S m q f (z) S m q f (z) Using (3.14) in (3.13) and coparing with (3.15), we obtain and For any complex number µ and after some calculation we have where Using a lemm(2.5)on (3.16) we have the required results.
Proof.Let we note that z∂ q S m q f (z) S m q f (z) From (3.18), we have z∂ q S m q f (z) S m q f (z) ) |γ| z∂ q S m q f (z) S m q f (z) − 1 = z∂ q S m q f (z) − S m q f (z) S m q f (z) Because from (3.8).

Theorem 3 . 4 .
If a function f (z) ∈ A has the form (1.1) satisfies the condition