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Journal of Complex Analysis
Volume 2013 (2013), Article ID 572718, 8 pages
Subclasses of Starlike Functions Associated with Fractional -Calculus Operators
1School of Advanced Sciences, VIT University, Vellore 632014, India
2Department of Mathematics, Presidency College (Autonomous), Chennai 600005, India
3Department of Mathematics, Dr. Ambedkar Government Arts College, Chennai 600039, India
Received 3 January 2013; Accepted 27 March 2013
Academic Editor: Vladislav Kravchenko
Copyright © 2013 G. Murugusundaramoorthy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Making use of fractional -calculus operators, we introduce a new subclass of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in . Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in .
1. Introduction and Preliminaries
Denote by the class of functions of the form which are analytic and univalent in the open disc and normalized by . Due to Silverman , denote by a subclass of 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 the -theory. Recently Purohit and Raina  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 as a product of factors by and in terms of basic analogue of the gamma function Due to Gasper and Rahman , the recurrence relation for -gamma function is given by and the binomial expansion is given by Further the -derivative and -integral of functions defined on the subset of are, respectively, given by It is interest to note that the familiar Pochhammer symbol. Due to Kim and Srivastava , 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 can be expressed as the -binomial given by (6) and the series is a single valued when and , therefore the function in (8) is single valued when , and .
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.
Definition 3. Under the hypothesis of Definition 2, the fractional derivative of order is defined by
With the aid of the above definitions, and their known extensions involving -differintegral operator we define the linear operator where where , and . Here in (10) represents, respectively, a fractional -integral of of order when and fractional -derivative of of order when .
In 1975, Silverman  studied two interesting subclasses of , namely, , the class of starlike function of order () if and , the class of convex function of order () if . Motivated by the earlier works of Goodman  and Rønning [6, 7] in this paper we define the following new subclass of -starlike functions of order based on the -fractional operator.
For different choices of we state some special cases of subclasses of as illustrated in the following examples.
Example 4. For , we let where , , , and .
Example 5. For and , we let where , and .
In this paper we determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in . Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in .
2. Characterization Properties of
We recall the following lemmas to prove our main results.
Lemma 6. If is a real number and is a complex number, then .
Lemma 7. If is a complex number and , are real numbers, then
Lemma 8. Let , , then a function if and only if where 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 . Using Lemma 7, it is enough to show that
That is, suppose , then by Lemma 7 and by choosing the values of on the positive real axis inequality (18) reduces to
Since , the previous inequality reduces to
Letting and by the mean value theorem we get desired inequality (17).
Conversely, let (17) hold; we will show that (13) is satisfied, and so . In view of Lemma 6, , it is enough to show that where Hence, one has and it is easy to show that , by the given condition (17), and the proof is complete.
Corollary 9. If , then , , , , where . Equality holds for the function .
For the sake of brevity we let unless otherwise stated.
Remark 10. First we show that the function is a decreasing function of for , . It follows that and it is sufficient to consider here the value , so that on using (5) we get The function is a decreasing function of if , and this gives . Multiplying the previous inequality both sides by provided ; we are at once lead to the inequality . Thus, is a decreasing function of for , .
Now by routine procedure using the techniques employed by Silverman  we can easily prove the following theorems.
Theorem 12 (extreme points). The extreme points of are and , for . Then if and only if it can be expressed in the form , , , where is defined in (24).
Theorem 13. Let the functions () defined by be in the classes (), respectively. Then the function is in the class , where ().
3. Neighbourhood Results
Theorem 14. If then , where is defined in (25).
Now we determine the neighborhood for the class which we define as follows. A function is said to be in the class if there exists a function such that
Theorem 15. If and then where is defined in (25).
4. Subordination Results
Now we recall the following results due to Wilf , which are very much needed for our study.
Definition 16 (subordination principle ). For analytic functions and with , is said to be subordinate to , denoted by , if there exists an analytic function such that , , and , for all .
Definition 17 (subordinating factor sequence). A sequence of complex numbers is said to be a subordinating sequence if, whenever , is regular, univalent, and convex in , we have
Lemma 18. The sequence is a subordinating factor sequence if and only if
Theorem 19. Let and be any function in the usual class of convex functions , then where , , , and The constant factor in (47) cannot be replaced by a larger number.
Proof. Let , and suppose that . Then
Thus, by Definition 17, the subordination result holds true if
is a subordinating factor sequence, with . In view of Lemma 18, this is equivalent to the following inequality:
By noting the fact that is increasing function for and in particular
therefore, for , 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.
The inequality (48) follows from (47) by taking Next we consider the function where , , , and is given by (25). Clearly . For this function (47) becomes It is easily verified that This shows that the constant cannot be replaced by any larger one.
5. Partial Sums
For a function given by (1) Silverman  and Silvia  investigated the partial sums and defined by We consider in this section partial sums of functions in the class and obtain sharp lower bounds for the ratios of real part of to and to .
Proof. By (61) it is not difficult to verify that Thus by Lemma 8 we have Setting it suffices to show that 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 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.
Proof. By setting the proof is analogous to that of Theorem 20, and we omit the details.
6. Integral Means
In , Silverman found that the function is often extremal over the family . He applied this function to resolve his integral means inequality, conjectured in  and settled in , that for all , , and . In , he also proved his conjecture for the subclasses 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 ). If the functions and are analytic in with , then for , and ,
Theorem 23. Suppose that , , , , , and is defined by where is defined in (25). Then for , , we have
Concluding Remarks. In fact, by suitably specializing the values of and , the results presented in this paper would find further applications for the class of univalent starlike functions with negative coefficients stated in Examples 4 and 5 of Section 1.
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