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## Partial Fractional Equations and their Applications

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Research Article | Open Access

Volume 2014 |Article ID 653917 | https://doi.org/10.1155/2014/653917

K. A. Selvakumaran, Sunil Dutt Purohit, Aydin Secer, "Majorization for a Class of Analytic Functions Defined by -Differentiation", Mathematical Problems in Engineering, vol. 2014, Article ID 653917, 5 pages, 2014. https://doi.org/10.1155/2014/653917

# Majorization for a Class of Analytic Functions Defined by -Differentiation

Accepted01 Jul 2014
Published10 Jul 2014

#### Abstract

We introduce a new class of multivalent analytic functions defined by using -differentiation and fractional -calculus operators. Further, we investigate majorization properties for functions belonging to this class. Also, we point out some new and known consequences of our main result.

#### 1. Introduction and Preliminaries

Let denote the class of functions of the form which are analytic and -valent in the open unit disk . For analytic functions and in , we say that the function is majorized by in (see ) and write if there exists a Schwarz function , analytic in , such that

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 , the definition (4) remains meaningful for as a convergent infinite product In view of the relation we observe that the -shifted factorial (4) reduces to the familiar Pochhammer symbol , where .

It may be noted that the -Gauss hypergeometric function (see Gasper and Rahman [2, p.3, eqn. (1.2.14)]) is defined by and as a special case of the above series for , we have Also, the -derivative and -integral of a function on a subset of are, respectively, given by (see [2, pp. 19–22]) Therefore, the -derivative of , where is a positive integer, is given by where and is called the -analogue of . As , we have . Here we list some relations satisfied by :

Recently, many authors have introduced new classes of analytic functions using -calculus operators. For some recent investigations on the classes of analytic functions defined by using -calculus operators and related topics, we refer the reader to  and the references cited therein. In the following, we define the fractional -calculus operators of a complex-valued function , which were recently studied by Purohit and Raina .

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 is given by The series is single valued when and (see for details , pp. 104–106); therefore, the function in (15) is single valued when , , and .

Definition 2 (fractional -derivative operator). The fractional -derivative operator of a function of order is defined by where is suitably constrained and the multiplicity of is removed as in Definition 1.

Definition 3 (extended fractional -derivative operator). Under the hypotheses of Definition 2, the fractional -derivative for a function of order is defined by where , and denotes the set of natural numbers.

Remark 4. It follows from Definition 2 that
Using , we define a -differintegral operator , as follows: where in (20) represents, respectively, a fractional -integral of of order when and a fractional -derivative of of order when . It is easy to see from (20) that

Definition 5. A function is said to be in the class of -valent functions of complex order in if and only if
It can be seen that, by specializing the parameters, the class reduces to many known subclasses of analytic functions. For instance, if then (1), the class of starlike functions of complex order (see ),(2), the class of convex functions of complex order (see ),(3), the class of -spiral-like functions (see ),(4), the class of starlike functions of order .

#### 2. Majorization Problem for the Class

We start by proving the following -analogue of the result given by Nehari in .

Lemma 6. If is analytic and bounded in , then

Proof. If is bounded in , then is also bounded in . Clearly vanishes when . Therefore, the function is regular when and also at all other points of . Furthermore, is bounded in . In fact, and for ; hence by maximum principle, throughout . Then from (26) we have

Theorem 7. Let the function be in the class and suppose that . If is majorized by in for , then where

Proof. Let Since , we have and where denotes the well known class of bounded analytic functions in , which satisfy the conditions (cf. ) It follows from (30) and (31) that In view of the identity (22), we have the following inequality from (33) by making some simple calculations: Since is majorized by in , there exists an analytic function such that and . Applying -differentiation with respect to and multiplying by , we have from (35) Using (22), in the above equation, we get Noting that is bounded in and using Lemma 6 we obtain Appling (34) and (38) in (37) we get where the function , defined by takes its maximum value at with given by (29). Furthermore, if where given by (29), then the function increases in the interval , so that does not exceed Therefore, from this fact, (39) gives inequality (28).

Letting , and in Theorem 7, we have the following.

Corollary 8 (see ). Let the function be analytic and univalent in the open unit disk and suppose that , the class of starlike functions of complex order . If is majorized by in , then where

For , Corollary 8 reduces to the following result.

Corollary 9. Let the function be analytic and univalent in the open unit disk and suppose that , the class of -spiral-like functions. If is majorized by in , then for .

Further setting , in Corollary 8 we get the following.

Corollary 10 (see ). Let the function be analytic and univalent in the open unit disk and suppose that is starlike in . If is majorized by in , then

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

1. T. H. MacGregor, “Majorization by univalent functions,” Duke Mathematical Journal, vol. 34, pp. 95–102, 1967.
2. 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
3. H. Aldweby and M. Darus, “A subclass of harmonic univalent functions associated with $q$-analogue of Dziok-Srivastava operator,” ISRN Mathematical Analysis, vol. 2013, Article ID 382312, 6 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
4. H. Aldweby and M. Darus, “On harmonic meromorphic functions associated with basic hypergeometric functions,” The Scientific World Journal, vol. 2013, Article ID 164287, 7 pages, 2013. View at: Publisher Site | Google Scholar
5. H. Aldweby and M. Darus, “Some subordination results on $q$-analogue of Ruscheweyh differential operator,” Abstract and Applied Analysis, vol. 2014, Article ID 958563, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
6. G. Murugusundaramoorthy, C. Selvaraj, and O. S. Babu, “Subclasses of starlike functions associated with fractional $q$-calculus operators,” Journal of Complex Analysis, vol. 2013, Article ID 572718, 8 pages, 2013. View at: Google Scholar | MathSciNet
7. G. Murugusundaramoorthy and T. Janani, “Meromorphic parabolic starlike functions associated with $q$-hypergeometric series,” ISRN Mathematical Analysis, vol. 2014, Article ID 923607, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
8. S. D. Purohit, “A new class of multivalently analytic functions associated with fractional q-calculus operators,” Fractional Differential Calculus, vol. 2, no. 2, pp. 129–138, 2012. View at: Google Scholar
9. 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.
10. S. D. Purohit and R. K. Raina, “Fractional q-calculus and certain subclass of univalent analytic functions,” Mathematica, vol. 55, no. 1, pp. 62–74, 2013. View at: Google Scholar
11. S. D. Purohit and R. K. Raina, “Some classes of analytic and multivalent functions associated with q-derivative operators,” Acta Universitatis Sapientiae, Mathematica, vol. 6, no. 1, 2014. View at: Google Scholar
12. S. D. Purohit and R. K. Raina, “On a subclass of p-valent analytic functions involving fractional q-calculus operators,” Kuwait Journal of Science, vol. 41, no. 3, 2014. View at: Google Scholar
13. K. A. Selvakumaran, S. D. Purohit, A. Secer, and M. Bayram, “Convexity of certain $q$-integral operators of $p$-valent functions,” Abstract and Applied Analysis, vol. 2014, Article ID 925902, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
14. M. A. Nasr and M. K. Aouf, “Starlike function of complex order,” The Journal of Natural Sciences and Mathematics, vol. 25, no. 1, pp. 1–12, 1985.
15. P. Wiatrowski, “The coefficients of a certain family of holomorphic functions,” Zeszyty Naukowe Uniwersytetu Lodzkiego, Nauki Matematyczno Przyrodnicze II, Zeszyt, no. 39, pp. 75–85, 1971. View at: Google Scholar
16. R. J. Libera, “Univalent $\alpha$-spiral functions,” Canadian Journal of Mathematics, vol. 19, pp. 449–456, 1967. View at: Google Scholar | MathSciNet
17. Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952. View at: MathSciNet
18. A. W. Goodman, Univalent Functions, Mariner, Tampa, Fla, USA, 1983. View at: MathSciNet
19. O. Altintaş, Ö. Özkan, and H. M. Srivastava, “Majorization by starlike functions of complex order,” Complex Variables: Theory and Application, vol. 46, no. 3, pp. 207–218, 2001. View at: Google Scholar | MathSciNet