Subclasses of Starlike Functions Associated with Fractional q-Calculus Operators

The fractional calculus operator has gained importance and popularly due to vast potential demonstrated applications in various fields of science, engineering and also in the geometric function theory.The fractional q-calculus operator is the extension of the ordinary fractional calculus in the qtheory. Recently Purohit and Raina [2] investigated applications of fractional q-calculus operator to define new classes of functions which are analytic in the open unit disc. We recall the definitions of fractional q-calculus operators of complex valued function f(z). The q-shifted factorial is defined for α, q ∈ C as a product of n factors by


Introduction and Preliminaries
Denote by A the class of functions of the form which are analytic and univalent in the open disc = { : | | < 1} and normalized by (0) = 0 = (0) − 1. Due to Silverman [1], denote by T a subclass of A consisting of functions of the form The fractional calculus operator has gained importance and popularly due to vast potential demonstrated applications in various fields of science, engineering and also in the geometric function theory. The fractional -calculus operator is the extension of the ordinary fractional calculus in thetheory. Recently Purohit and Raina [2] investigated applications of fractional -calculus operator to define new classes of functions which are analytic in the open unit disc. We recall the definitions of fractional -calculus operators of complex valued function ( ).
The -shifted factorial is defined for , ∈ C as a product of factors by and in terms of basic analogue of the gamma function Due to Gasper and Rahman [3], the recurrence relation for -gamma function is given by and the binomial expansion is given by 2

Journal of Complex Analysis
Further the -derivative and -integral of functions defined on the subset of C are, respectively, given by It is interest to note that lim → 1 − (( ; ) /(1 − ) ) = ( ) = ( + 1) ⋅ ⋅ ⋅ ( + − 1) the familiar Pochhammer symbol. Due to Kim and Srivastava [4], we recall the following definitions of fractional -integral and fractional -derivative operators, which are very much useful for our study. Definition 1. Let the function ( ) be analytic in a simply connected region of the -plane containing the origin. The fractional -integral of of order is defined by where ( − ) −1 can be expressed as the -binomial given by (6) and the series 1 Φ 0 [ ; −; , ] is a single valued when | arg( )| < and | | < 1, therefore the function ( − ) −1 in (8) is single valued when | arg(− / )| < , | | < 1 and | arg( )| < .
Definition 2. The fractional -derivative operator of order is defined for a function ( ) by where the function ( ) is constrained, and the multiplicity of the function ( − ) − is removed as in Definition 1.
Journal of Complex Analysis 3 In this paper we determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in M ( , , ). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in M ( , , ).

Characterization Properties of M ( , , )
We recall the following lemmas to prove our main results.

Lemma 6. If is a real number and is a complex number,
If is a complex number and , are real numbers, then where Φ ( , ) is given by (12).
Proof. Let a function of the form (2) in satisfy the condition (17). We will show that (13) is satisfied, and so ∈ M ( , , ). Using Lemma 7, it is enough to show that That is, suppose ∈ M ( , , ), then by Lemma 7 and by choosing the values of on the positive real axis inequality (18) reduces to Since Re(− ) ≥ − 0 = −1, the previous inequality reduces to Letting → 1 − and by the mean value theorem we get desired inequality (17). Conversely, let (17) hold; we will show that (13) is satisfied, and so ∈ M ( , , ). In view of Lemma where Hence, one has Journal of Complex Analysis and it is easy to show that − > 0, by the given condition (17), and the proof is complete.
unless otherwise stated.
Now by routine procedure using the techniques employed by Silverman [1] we can easily prove the following theorems.

Neighbourhood Results
In this section, we discuss neighbourhood results of the class M ( , , ). Following [8,9], we define the -neighbourhood of function ( ) ∈ T by Particularly for the identity function ( ) = , we have so that .

Subordination Results
Now we recall the following results due to Wilf [10], which are very much needed for our study.
Definition 17 (subordinating factor sequence). A sequence { } ∞ =1 of complex numbers is said to be a subordinating sequence if, whenever ( ) = ∑ ∞ =1 , 1 = 1 is regular, univalent, and convex in , we have where we have also made use of the assertion (17) of Lemma 8. This evidently proves the inequality (51) and hence also the subordination result (47) asserted by Lemma 8.

Partial Sums
For a function ∈ given by (1) Silverman [12] and Silvia [13] investigated the partial sums 1 and defined by We consider in this section partial sums of functions in the class M ( , , ) and obtain sharp lower bounds for the ratios of real part of to ( ) and to .
Theorem 20. Let a function of the form (1) belong to the class M ( , , ) and satisfy the condition (17). Then where Proof. By (61) it is not difficult to verify that Thus by Lemma 8 we have Setting it suffices to show that R ( ( )) ≥ 0, ∈ .
Applying (63), we find that which readily yields the assertion (59) of Theorem 20. In order to see that gives the sharp result, we observe that for = / we have Similarly, if we take Journal of Complex Analysis 7 and making use of (63), we can deduce that which leads us immediately to the assertion (60) of Theorem 20. The bound in (60) is sharp for each ∈ with the extremal function given by (67), and the proof is complete.
Theorem 21. Let a function of the form (1) belong to the class M ( , , ) and satisfy the condition (17). Then where C is defined by (61) Proof. By setting the proof is analogous to that of Theorem 20, and we omit the details.

Integral Means
In [1], Silverman found that the function 2 ( ) = − ( 2 /2) is often extremal over the family T. He applied this function to resolve his integral means inequality, conjectured in [14] and settled in [15], that for all ∈ T, > 0, and 0 < < 1. In [15], he also proved his conjecture for the subclasses T * ( ) the class of starlike functions and ( ) the class of convex functions with negative coefficients. We recall the following lemma to prove our result on integral means inequality.
Lemma 22 (see [11]). If the functions and are analytic in with ≺ , then for > 0, and 0 < < 1, Applying Lemmas 22 and 8 and Theorem 12, we prove the Silverman's conjecture for the functions in the family M ( , , ) by using known procedures.