Subordination problems for a new class of Bazilevič functions associated with k-symmetric points and fractional q-calculus operators

1 School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China 2 Department of Mathematics, R. M. K College of Engineering and Technology, Puduvoyal 601206, Tamil Nadu, India 3 Department of Mathematics, University College of Engineering Tindivanam, Anna University (Chennai), Tindivanam 604001, Tamil Nadu, India 4 Department of HEAS (Mathematics), Rajasthan Technical University, Kota 324010, Rajasthan, India


Introduction and preliminaries
The hypothesis of fractional calculus, which is primarily due to its proven developments in many branches of science and engineering over the past three decades or so, has gained considerable prominence and recognition in recent times. Most of the theory of fractional calculus is specifically based upon the familiar Riemann-Liouville fractional derivative (or integral). The fractional q-calculus is the extension of the ordinary fractional calculus in the q-theory. Recently, there was notable increase of articles written in the area of the q-calculus due to significant applications of the q-calculus in mathematics, statistics and physics. For more details, interested readers may refer to the books of [1, 2, 4-6, 8, 9, 14] on the subject.
Indeed the fourth author of this article with Raina in [11] have used the fractional q-calculus operators in investigating certain classes of functions which are analytic in the open disk D. Recently, many authors have introduced new classes of analytic functions using q-calculus operators. For some recent investigations on the classes of analytic functions defined by using q-calculus operators and related topics, we refer the reader to [12,13,17,18] and the references cited therein. In the present paper, we aim at introducing a new class of Bazilevič functions involving the fractional q-calculus operators, which is analytic in the open unit disk. Certain interesting subordination results are also derived for the functions belonging to this class.
We first give various definitions and notations in q-calculus which are useful to understand the subject of this paper.
For any complex number α, the q-shifted factorials are defined as pα; qq 0 1, pα; qq n n¡1 ¹ k0 p1 ¡ αq k q, n N, (1.1) and in terms of the basic analogue of the gamma function where the q-gamma function is defined by If |q| 1, the definition (1.1) remains meaningful for n V as a convergent infinite product In view of the relation lim qÑ1 ¡ pq α ; qq n p1 ¡ qq n pαq n , we observe that the q-shifted factorial (1.1) reduces to the familiar Pochhammer symbol pαq n , where pαq n αpα 1q ¤ ¤ ¤ pα n ¡ 1q.
Also, the q-derivative and q-integral of a function on a subset of C are, respectively, given by (see [6] pp. [19][20][21][22]) and Therefore, the q-derivative of f pzq z n , where n is a positive integer is given by D q z n z n ¡ pzqq n p1 ¡ qqz rns q z n¡1 , where rns q 1 ¡ q n 1 ¡ q q n¡1 ¤ ¤ ¤ 1 and is called the q-analogue of n. As q Ñ 1, we have rns q q n¡1 ¤ ¤ ¤ 1 Ñ 1 ¤ ¤ ¤ 1 n. The q-derivative of f pzq ln z is given by p1 ¡ qqz . We now define the fractional q-calculus operators of a complex-valued function f pzq, which were recently studied by Purohit and Raina [11].
where f pzq is suitably constrained and the multiplicity of pz ¡ tqq ¡δ is removed as in Definition 1.1. where m ¡ 1 ¤ δ 1, m N 0 N t0u, and N denotes the set of natural numbers.
2. The class B m q,k pλ, δ, γ; φq Let Hpa, nq denote the class of functions f pzq of the form f pzq a a n z n a n 1 z n 1 ¤ ¤ ¤ , pz Dq, (2.1) which are analytic in the unit disk D tz C : |z| 1u. In particular, let A be the subclass of 3) The function f is said to be starlike with respect to k-symmetric points if it satisfies We denote by S pkq s the subclass of A consisting of all functions starlike with respect to k-symmetric points in D. The class S p2q s was introduced and studied by Sakaguchi [15]. We also note that different subclasses of S pkq s can be obtained by replacing condition (2.4) by z f I pzq f k pzq hpzq, where hpzq is a given convex function, with hp0q 1 and thpzqu ¡ 0.
Using D δ q,z , we define a fractional q-differintegral operator Ω δ q,z : A ÝÑ A, as follows: Γ q p2 ¡ δqΓ q pn 1q Γ q p2qΓ q pn 1 ¡ δq a n z n , q,z f pzq in (2.5) represents, respectively, a fractional q-integral of f pzq of order δ when ¡V δ 0 and a fractional q-derivative of f pzq of order δ when 0 ¤ δ 2. Here we note that Ω 0 q,z f pzq f pzq.
If f pzq is given by (2.2), then by (2.6), we have It can be seen that, by specializing the parameters the operator D δ,m q,λ reduces to many known and new integral and differential operators. In particular, when δ 0 the operator D δ,m q,λ reduces to the operator introduced by Al-Oboudi [3] and for δ 0, λ 1 it reduces to the operator introduced by Sȃlȃgean [16]. Throughout this paper, we assume that f m q,k pλ, δ; zq 1 k k¡1 j0 ε ¡j k pD δ,m q,λ f pε j k zqq z ¤ ¤ ¤ , pf Aq. (2.7) Clearly, for k 1, we have f m q,1 pλ, δ; zq D δ,m q,λ f pzq.
Let P denote the class of analytic functions hpzq with hp0q 1, which are convex and univalent in D and for which thpzqu ¡ 0, pz Dq.
It is easy to see to verify the following relations: If q Ñ 1 ¡ , then 3. B 0 q,1 pλ, δ, 1; 1 z 1¡z q K; 4. B 0 q,1 pλ, δ, γ; 1 z 1¡z q Bpγq, the class of Bazilevič functions of type γ. Now, we derive some sufficient conditions for functions belonging to the class B m q,k pλ, δ, γ; φq. In order to prove our results we need the following results.
Lemma 2.1. p [10], see also [7]q Let h be convex in D, with hp0q a, β $ 0 and pβq ¥ 0. The function φ is convex and is the best (a, n)-dominant.
Lemma 2.2. p [10], see also [19]q Let h be starlike in D, with hp0q 0. If g Hpa, nq satisfies zg I pzq hpzq, then gpzq φpzq a n ¡1 The function φ is convex and is the best (a, n)-dominant.
3. Subordination problems for the class B m q,k pλ, δ, γ; φq In this section, we establish some sufficient conditions of subordination for analytic functions defined above involving the q-differential operator. Since h is convex, we have ¢ p 1 pzq z p I 1 pzq p 1 pzq 1 ¡ 0 which implies pp 1 pzqq ¡ 0 ( [10], Theorem 3.2a) and therefore, Qpzq is convex and univalent. We now set Ppzq p 2 pzq. Then Ppzq Hp1, 1q with Ppzq $ 0 in D. By  and Q is the best dominant of (3.5). Since thpzqu ¡ 0 and Qpzq hpzq we also have tQpzqu ¡ 0. Hence, the univalence of Q implies the univalence of qpzq Qpzq, and p 2 pzq Ppzq Qpzq q 2 pzq, which implies that ppzq qpzq. Since Q is the best dominant of (3.5), we deduce that qpzq is the best dominant of (3.2). This completes the proof.