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Journal of Complex Analysis
Volume 2013 (2013), Article ID 572718, 8 pages
http://dx.doi.org/10.1155/2013/572718
Research Article

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.

Abstract

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 [1], 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 [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 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 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 [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 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 [1] 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 [5] and Rønning [6, 7] in this paper we define the following new subclass of -starlike functions of order based on the -fractional operator.

For , , , and , we let be the subclass of consisting of functions of the form (2) and satisfying the analytic criterion , and is given by (11).

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 [1] we can easily prove the following theorems.

Theorem 11. Let the function defined by (2) belong to , then Equalities are sharp for the function , where is obtained from (25).

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

In this section, we discuss neighbourhood results of the class . Following [8, 9], we define the -neighbourhood of function by Particularly for the identity function , we have

Theorem 14. If then , where is defined in (25).

Proof. For , Lemma 8 immediately yields so that On the other hand, we find from (17) and (36) that which, in view of the definition (33), proves Theorem 14.

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).

Proof. Suppose that ; we then find from (32) that which yields Next, since , we have so that provided that is given precisely by (39). Thus for given by (39), completes the proof.

4. Subordination Results

Now we recall the following results due to Wilf [10], which are very much needed for our study.

Definition 16 (subordination principle [11]). 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 [12] and Silvia [13] 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 .

Theorem 20. Let a function of the form (1) belong to the class 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 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.

Theorem 21. Let a function of the form (1) belong to the class and satisfy the condition (17). Then where is defined by (61)

Proof. By setting the proof is analogous to that of Theorem 20, and we omit the details.

6. Integral Means

In [1], Silverman found that the function is often extremal over the family . He applied this function to resolve his integral means inequality, conjectured in [14] and settled in [15], that for all , , and . In [15], 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 [11]). If the functions and are analytic in with , then for , and ,

Applying Lemmas 22 and 8 and Theorem 12, we prove the Silverman's conjecture for the functions in the family by using known procedures.

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.

References

  1. H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, pp. 109–116, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. D. Purohit and R. K. Raina, “Certain subclasses of analytic functions associated with fractional q-calculus operators,” Mathematica Scandinavica, vol. 109, no. 1, pp. 55–70, 2011. View at MathSciNet
  3. G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990. View at MathSciNet
  4. Y. C. Kim and H. M. Srivastava, “Fractional integral and other linear operators associated with the Gaussian hypergeometric function,” Complex Variables. Theory and Application, vol. 34, no. 3, pp. 293–312, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. W. Goodman, “On uniformly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 364–370, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Rønning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam, and H. Silverman, “Subclasses of uniformly convex and uniformly starlike functions,” Mathematica Japonica, vol. 42, no. 3, pp. 517–522, 1995. View at MathSciNet
  8. A. W. Goodman, “Univalent functions and nonanalytic curves,” Proceedings of the American Mathematical Society, vol. 8, pp. 598–601, 1957. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. S. Wilf, “Subordinating factor sequences for convex maps of the unit circle,” Proceedings of the American Mathematical Society, vol. 12, pp. 689–693, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathematical Society, vol. 23, no. 1, pp. 481–519, 1925. View at Publisher · View at Google Scholar · View at MathSciNet
  12. H. Silverman, “Partial sums of starlike and convex functions,” Journal of Mathematical Analysis and Applications, vol. 209, no. 1, pp. 221–227, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. M. Silvia, “On partial sums of convex functions of order α,” Houston Journal of Mathematics, vol. 11, no. 3, pp. 397–404, 1985. View at Zentralblatt MATH · View at MathSciNet
  14. H. Silverman, “A survey with open problems on univalent functions whose coefficients are negative,” The Rocky Mountain Journal of Mathematics, vol. 21, no. 3, pp. 1099–1125, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. Silverman, “Integral means for univalent functions with negative coefficients,” Houston Journal of Mathematics, vol. 23, no. 1, pp. 169–174, 1997. View at Zentralblatt MATH · View at MathSciNet