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Fractal Fract 2019, 3(1), 5; https://doi.org/10.3390/fractalfract3010005
On q-Uniformly Mocanu Functions
Department of Mathematics, COMSATS, University Islamabad, Islamabad 44000, Pakistan
Author to whom correspondence should be addressed.
Received: 28 January 2019 / Accepted: 10 February 2019 / Published: 11 February 2019
Let f be analytic in open unit disc with and . The q-derivative of f is defined by: where is a q-geometric subset of . Using operator , q-analogue class , k-uniformly Mocanu functions are defined as: For and , reduces to of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.
Keywords:q-calculus; q-starlike; uniformly convex; subordination; Mocanu functions; q-Ruscheweyh derivative
Let A denote the class of functions f that are analytic in the open unit disc E and are also normalized by the conditions , . Let f is said to be subordinate to g (written as ), if there exists a Schwartz function such that
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].
We recall here some basic concepts from q-calculus for which we refer to [5,6,7,8,9,10,11,12,13,14,15,16] and the references therein.
A subset is called q-geometric, if whenever , and it contains all the geometric sequences .
The q-derivative of a function is defined by:and
Under this definition, we have the following rules for q-derivative
Let and be defined on a q-geometric set such that q-derivatives of and exist for all . Then, for complex numbers, we have:
Let be the class of functions analytic in E and satisfying:
It is known  that implies that , where ≺ denotes subordination, and from this, it easily follows that
Now, we have:
We can write (3) as:
As a special case, we note that:which is the class of starlike functions denoted as .
Furthermore, for , we obtain the class of q-starlike functions introduced and studied in .
Let and , . Then,if and only if, for
Selecting special values of parameters and k and letting , we obtain a number of known classes of analytic functions; see [5,9,18,19,20,21]. We list some of these as follows:
Throughout this paper, we shall assume that and unless otherwise mentioned.
2. Preliminary Results
. Let be analytic with . If attains its maximum value on the circle at a point , then we have:
. Let and . Let be analytic in E with .
3. Main Results
Let be analytic in E with Let, for ,
Then, is subordinate to , that is, in
Let . It can easily be seen that is analytic in E and . We shall show that for all We suppose on the contrary that there exists a such that
Now, by Lemma 1, and we use it in (6) for:
This is a contradiction, and hence, for all . This proves that:□
We apply Theorem 1 to have the following results.
Let , , and . Then, from Theorem 1, it follows that:which implies , and so, in E.
For , let Then, in E.
The proof is immediate when we take in Theorem 1.
As a special case, when implies in
Using a similar technique, we can prove the following results.
Let , and let be analytic in E with
We can easily deduce some special cases of Theorem 2 as given below.
Let in (8). Then:implies:
As a special case of this corollary, we observe that when we choose , and let
Let and Then:
This gives us:
Now, using Lemma 2 together with Theorem 2 when we obtain the result that:
In (8), if we take and , then:implies
Furthermore, with and in (8), it follows that:implies
Next, we prove the following:
Let be analytic in E with Let:where , and c are positive real. Then, in
We shall follow the same procedure to prove this result as was used in Theorem 1. Let . Clearly, , and is analytic. We prove that is a Schwartz function, that is . Suppose on the contrary that there exists such that .
Now, with some computations, we have:
We apply Lemma 1 to have , and note that:and:
Using (10), (11), (12), and (13), we get a contradiction to the given hypothesis (9), when we assume for some . Hence for all and:
This completes the proof. □
In order to develop some applications of Theorem 3, we need the following.
Let the operator be defined as:where:and:
This series is absolutely convergent in E, and * denotes convolution. The operator is called the q-Ruscheweyh derivative of order n; see .
It can easily be seen that and
The relation (14) can be expressed as:
Furthermore,which is called the Ruscheweyh derivative of order n; see .
Let . Then, f is said to belong to the class if and only if,
The following identity can easily be obtained:
We now take in relation (9) of Theorem 3 to have:
Let denote the q-Ruscheweyh derivative of order n for Let:
Let p be analytic in E with , and let:
Using identity (15) and some computation, we have:
Now, the required result follows immediately from Theorem 3. □
In Theorem 4, we take . Then, it gives us:
When , and we have:
Let , and let:
The integral operator defined in (16) is known as the q-Bernardi integral operator When (16) reduces to the well-known Bernardi operator; see .
We now apply Theorem 3, and it follows that:
That is, □
As a special case, when then , and then, defined by 17, belongs to in
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.
Conceptualization, K.I.N.; formal analysis, K.I.N.; investigation, R.S.B. and K.I.N.; methodology, R.S.B. and K.I.N.; supervision, K.I.N.; validation, R.S.B. and K.I.N.; writing, original draft, R.S.B. and K.I.N.; writing, review and editing, K.I.N.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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