Convexity of Certain q-Integral Operators of p-Valent Functions

and Applied Analysis 3 where f(z) is analytic in a simply connected region of the zplane containing the origin and the q-binomial function (z − tq)δ−1 is given by (z − tq) δ−1 = z δ−1 1 Φ0 [q −δ+1 ; −; q, tq δ z ] . (19) The series 1 Φ0[δ; −; q, z] is single valued when | arg(z)| < π and |z| < 1 (see for details [2], pp. 104–106); therefore, the function (z − tq) δ−1 in (18) is single valued when | arg(−tq/z)| < π, |tq/z| < 1, and | arg(z)| < π. Definition 2 (fractional q-derivative operator). The fractional q-derivative operator D q,z of a function f(z) of order δ is defined by D δ q,z f (z) ≡ Dq,z I 1−δ q,z f (z) = 1 Γq (1 − δ)


Introduction and Preliminaries
The subject of fractional calculus has gained noticeable importance and popularity due to its established applications in many fields of science and engineering during the past three decades or so.Much of the theory of fractional calculus is based upon the familiar Riemann-Liouville fractional derivative (or integral).The fractional -calculus is the extension of the ordinary fractional calculus in the theory.Recently, there was a significant increase of activity in the area of the -calculus due to applications of the -calculus in mathematics, statistics, and physics.For more details, one may refer to the books [1][2][3][4] on the subject.Recently, Purohit and Raina [5][6][7] have added one more dimension to this study by introducing certain subclasses of functions which are analytic in the open disk U, by using fractional -calculus.Purohit [8] also studied similar work and considered new classes of multivalently analytic functions in the open unit disk.
The aim of this paper is to consider a linear multiplier fractional -differintegral operator and to define certain new subclasses of functions which are -valent and analytic in the open unit disk.The results derived include convexity properties of these -integral operators on some classes of analytic functions.Special cases of the main results are also mentioned.
We denote by S *  () the class of all such functions.On the other hand, a function  ∈ A  is said to be in the class C  () of -valently convex of order  (0 ≤  < ) if and only if Note that S *  (0) = S *  and C  (0) =   are, respectively, the classes of -valently starlike and -valently convex functions in U. Also, we note that S * 1 (0) = S * and C 1 (0) = C are, respectively, the usual classes of starlike and convex functions in U. A function  ∈ A  is said to be in the class US  (, ) ( For uniformly starlike and uniformly convex functions we refer to the papers [9][10][11].Note that US 1 (, ) = UST(, ) and UC 1 (, ) = UCV(, ), where the classes UST(, ) and UCV(, ) are, respectively, the classes of -uniformly starlike of order  (0 ≤  < 1) and -uniformly convex of order  (0 ≤  < 1) studied in [12].
For the convenience of the reader, we now give some basic definitions and related details of -calculus which are used in the sequel.
For any complex number  the -shifted factorials are defined as and in terms of the basic analogue of the gamma function where the -gamma function is defined by If || < 1, the definition (6) remains meaningful for  = ∞ as a convergent infinite product: In view of the relation lim we observe that the -shifted factorial (6) reduces to the familiar Pochhammer symbol ()  , where ()  = ( + 1) ⋅ ⋅ ⋅ ( +  − 1).Also, the -derivative and -integral of a function on a subset of C are, respectively, given by (see [2] pp.[19][20][21][22]) Therefore, the -derivative of () =   , where  is a positive integer, is given by where and is called the -analogue of .As  → 1, we have The -analogues to the function classes S *  (), C  (), US  (, ), and UC  (, ) are given as follows.
A function  ∈ A  is said to be in the class S * , () of -valently starlike with respect to -differentiation of order Also, a function  ∈ A  is said to be in the class C , () of -valently convex with respect to -differentiation of order On the other hand, a function  ∈ A  is said to be in the class US , (, ) of -uniformly -valent starlike with respect to -differentiation of order  (−1 ≤  < ) if it satisfies Furthermore, a function  ∈ A  is said to be in the class UC , (, ) of -uniformly -valent convex with respect to -differentiation of order  (−1 ≤  < ) if it satisfies In the following, we define the fractional -calculus operators of a complex-valued function (), which were recently studied by Purohit and Raina [5].
Definition 1 (fractional -integral operator).The fractional integral operator   , of a function () of order  is defined by where () is analytic in a simply connected region of the plane containing the origin and the -binomial function ( − ) −1 is given by The series where () is suitably constrained and the multiplicity of ( − ) − is removed as in Definition 1.

Definition 3 (extended fractional 𝑞-derivative operator).
Under the hypotheses of Definition 2, the fractional derivative for a function () of order  is defined by where  − 1 ≤  < 1,  ∈ N 0 = N ∪ {0}, and N denotes the set of natural numbers.
We now define a linear multiplier fractional differintegral operator D , ,, as follows: ( If () ∈ A  is given by ( 1), then by (24) we have It can be seen that, by specializing the parameters, the operator D , ,, reduces to many known and new integral and differential operators.In particular, when  = 0,  = 1, and  → 1 the operator D , ,, reduces to the operator introduced by AL-Oboudi [13] and if  = 0,  = 1,  = 1, and  → 1 it reduces to the operator introduced by Sǎlǎgean [14].
By using the operator D and   () : A   → A  is defined as where D ,  ,,   () is given by (24).
In this paper, we obtain the order of convexity with respect to -differentiation of the -integral operators   () and   () on the classes US , ,, (, ) and UC , ,, (, ).As special cases, the order of convexity of the operators ∫  0 (()/)   and ∫  0 (  ())   is also given.Proof.From (28), we observe that   () ∈ A  .On the other hand, it is easy to verify that

Convexity of the Operator 𝐹 𝑞
This completes the proof.