On q-Uniformly Mocanu Functions

Abstract: Let f be analytic in open unit disc E = {z : |z| < 1} with f (0) = 0 and f (0) = 1. The q-derivative of f is defined by: Dq f (z) = f (z)− f (qz) (1−q)z , q ∈ (0, 1), z ∈ B − {0}, where B is a q-geometric subset of C. Using operator Dq, q-analogue class k−UMq(α, β), k-uniformly Mocanu functions are defined as: For k = 0 and q → 1−, k− reduces to M(α) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.


Introduction
Let A denote the class of functions f that are analytic in the open unit disc E and are also normalized by the conditions f (0) = 0, f (0) = 1.Let f , g ∈ A. f is said to be subordinate to g (written as f ≺ g), if there exists a Schwartz function w(z) such that f (z) = g(w(z)).
q-calculus is ordinary calculus without a limit, and it has been used recently by many researchers in the field of geometric function theory.q-derivatives and q-integrals play an important and significant role in the study of quantum groups and q-deformed super-algebras, the study of fractal and multi-fractal measures and in chaotic dynamical systems.The name q-calculus also appears in other contexts; see [1,2].The most sophisticated tool that derives functions in non-integer order is the long-known fractional calculus; see [1][2][3][4].
A subset β ⊂ C is called q-geometric, if zq ∈ β, whenever z ∈ B, and it contains all the geometric sequences {zq m } ∞ 0 .The q-derivative D q of a function f ∈ A is defined by: and D q f (0) = f (0).Under this definition, we have the following rules for q-derivative D q . (i).
, where [m, q] = 1−q m 1−q .Let f (z) and g(z) be defined on a q-geometric set B ⊂ C such that q-derivatives of f (z) and g(z) exist for all z ∈ B. Then, for a, b complex numbers, we have: , g(z).g(qz) = 0.
As a special case, we note that: which is the class of starlike functions denoted as S * (α).Furthermore, for α = 0, we obtain the class S * q of q-starlike functions introduced and studied in [10].

Main Results
Theorem 1.Let p(z) be analytic in E with p(0 Then, p(z) is subordinate to . It can easily be seen that φ(z) is analytic in E and φ(0) = 0. We shall show that |φ(z)| < 1 for all z ∈ E. We suppose on the contrary that there exists a z 0 ∈ E such that |φ(z 0 )| = 1. Then: Now, by Lemma 1, and we use it in (6) for: From ( 6), (7), and choosing θ = π, we have: This is a contradiction, and hence, |φ(z)| < 1 for all z ∈ E. This proves that: We apply Theorem 1 to have the following results.
The proof is immediate when we take p(z) = f (z) z in Theorem 1.
As a special case, when Using a similar technique, we can prove the following results.
Proof.We shall follow the same procedure to prove this result as was used in Theorem 1.Let p(z) = 1 1−qφ(z) .Clearly, φ(0) = 0, and φ(z) is analytic.We prove that φ(z) is a Schwartz function, that is |φ(z)| < 1, ∀z ∈ E. Suppose on the contrary that there exists z Now, with some computations, we have: We apply Lemma 1 to have z and note that: and: Using ( 10), ( 11), (12), and ( 13), we get a contradiction to the given hypothesis ( 9), when we assume |φ(z • )| = 1 for some z • ∈ E. Hence |φ(z)| < 1 for all z ∈ E and: This completes the proof.
In order to develop some applications of Theorem 3, we need the following.Let the operator D n q : A → A be defined as: where: a m z m , and: This series is absolutely convergent in E, and * denotes convolution.The operator D n q is called the q-Ruscheweyh derivative of order n; see [25].
It can easily be seen that D • q f (z) = f (z) and D q f (z) = zD q f (z).The relation (14) can be expressed as: Furthermore, which is called the Ruscheweyh derivative of order n; see [25].
Let f ∈ A. Then, f is said to belong to the class S * q (n, α), if and only if, D n q f ∈ S * q (α), z ∈ E.
The following identity can easily be obtained: We now take p(z) = zD q (D n q f (z)) in relation (9) of Theorem 3 to have: Theorem 4. Let D n q f = F n denote the q-Ruscheweyh derivative of order n for f ∈ A. Let: Then: That is, f ∈ S * q (n, α), α = 1 1+q .
Proof.Let p be analytic in E with p(0) = 0, and let: Using identity (15) and some computation, we have: Now, the required result follows immediately from Theorem 3.
We now apply Theorem 3, and it follows that: That is, F c ∈ S * q ( 1 1+q ).
As a special case, when q → 1 − , then f ∈ K − UST( 12 ), and then, F c , defined by 17, belongs to S * ( 12 ) in E.

Conclusions
In this paper, we have used q-calculus, conic domains, and subordination to define and study some new subclasses involving Mocanu functions.Some interesting inclusion and subordination properties of these new classes have been derived.The q-analogue of the Ruscheweyh derivative has been used to obtain a new subordination result for q-Mocanu functions.Some special cases have been discussed as applications of our main results.The technique and ideas of this paper may stimulate further research in this dynamic field.