/ / Article

Research Article | Open Access

Volume 2017 |Article ID 7156738 | 5 pages | https://doi.org/10.1155/2017/7156738

# On Fekete-Szegö Problems for Certain Subclasses Defined by -Derivative

Accepted02 Aug 2017
Published07 Sep 2017

#### Abstract

We derive the Fekete-Szegö theorem for new subclasses of analytic functions which are -analogue of well-known classes introduced before.

#### 1. Introduction

Denote by the class of all analytic functions of the form in the open unit disk

For two analytic functions and in , the subordination between them is written as . Frankly, the function is subordinate to if there is a Schwarz function with , , for all , such that for all . Note that, if is univalent, then if and only if and .

In [1, 2], Jackson defined the -derivative operator of a function as follows: and . In case for is a positive integer, the -derivative of is given by As and , we have

Quite a number of great mathematicians studied the concepts of -derivative, for example, by Gasper and Rahman , Aral et al. , Li et al. , and many others (see ).

Making use of the -derivative, we define the subclasses and of the class for byThese classes are also studied and introduced by Seoudy and Aouf .

Noting that where and are, respectively, the classes of starlike of order and convex of order in ([17, 18]).

Next, we state the -analogue of Ruscheweyh operator given by Aldweby and Darus  that will be used throughout.

Definition 1 (see ). Let . Denote by the -analogue of Ruscheweyh operator defined by where given by is as follows:

From the definition we observe that if , we have where is Ruscheweyh differential operator defined in .

Using the principle of subordination and -derivative, we define the classes of -starlike and -convex analytic functions as follows.

Definition 2. For and , the class which consists of all analytic functions satisfies

Definition 3. For and , the class which consists of all analytic functions satisfies

To prove our results, we need the following.

Lemma 4 (see ). If of positive real part is in and is a complex number, then The result is sharp given by

Lemma 5 (see ). If is a function with positive real part, then

#### 2. Main Results

Now is our theorem using similar methods studied by Seoudy and Aouf in .

Theorem 6. Let . If given by (1) is in the class and is a complex number, then The result is sharp.

Proof. If , then there is a function in with and in such that Define the function by Since is a Schwarz function, immediately and . Let Then from (16), (17), and (18), obtain Since we have From the last equation and (18), we obtain A simple computation in (18) and knowing that , we obtain Then, from last equation and (18), we see that or equivalently, we have Therefore where By an application of Lemma 4, our result follows. Again by Lemma 4, the equality in (15) is gained for Thus Theorem 6 is complete.

Similarly, we can prove for the class . We omit the proofs.

Theorem 7. Let . If given by (1) is in the class and is a complex number, then The result is sharp.

Taking in Theorem 6, we have the corollary for the class as follows.

Corollary 8. Let . If given by (1) is in the class and is a complex number, then The result is sharp.

Taking and in Theorem 6, we obtain the following.

Corollary 9. Let , . If given by (1) is in the class and is a complex number, then

By using Lemma 4, we have the following theorem.

Theorem 10. Let with and . Let Let given by (1) be in the class . Then

Proof. First, let Now, let ; then using the above calculation, we obtain Finally, if , then

Similarly, we can prove for the class as follows.

Theorem 11. Let with and . Let If given by (1) is in the class , then

Taking in Theorem 10, we obtain next result for the class .

Corollary 12. Let with and . Let If given by (1) is in the class , then

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The work here is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.

1. F. H. Jackson, “On q- definite integrals,” The Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910. View at: Google Scholar
2. F. H. Jackson, “On q-functions and a certain difference operator,” Transactions of the Royal Society of Edinburgh, vol. 46, no. 2, pp. 253–281, 1909. View at: Publisher Site | Google Scholar
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. A. Aral, V. Gupta, and R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, NY, USA, 2013. View at: Publisher Site | MathSciNet
5. X. Li, Z. Han, S. Sun, and L. Sun, “Eigenvalue problems of fractional q-difference equations with generalized p-Laplacian,” Applied Mathematics Letters, vol. 57, pp. 46–53, 2016. View at: Publisher Site | Google Scholar | MathSciNet
6. 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
7. H. Al Dweby 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
8. 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
9. 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, pp. 5–23, 2014. View at: Google Scholar | MathSciNet
10. 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
11. T. M. Seoudy and M. K. Aouf, “Convolution properties for certain classes of analytic functions defined by q-derivative operator,” Abstract and Applied Analysis, vol. 2014, Article ID 846719, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
12. B. Wongsaijai and N. Sukantamala, “Applications of fractional q-calculus to certain subclass of analytic p-valent functions with negative coefficients,” Abstract and Applied Analysis, vol. 2015, Article ID 273236, 12 pages, 2015. View at: Publisher Site | Google Scholar | MathSciNet
13. U. A. Ezeafulukwe and M. Darus, “A note on q-calculus,” Fasciculi Mathematici, no. 55, pp. 53–63, 2015. View at: Google Scholar | MathSciNet
14. U. A. Ezeafulukwe and M. Darus, “Certain properties of q-hypergeometric functions,” International Journal of Mathematics and Mathematical Sciences, vol. 2015, Article ID 489218, 9 pages, 2015. View at: Publisher Site | Google Scholar | MathSciNet
15. H. Aldweby and M. Darus, “Coefficient estimates of classes of Q-starlike and Q-convex functions,” Advanced Studies in Contemporary Mathematics, vol. 26, no. 1, pp. 21–26, 2016. View at: Google Scholar
16. T. M. Seoudy and M. K. Aouf, “Coefficient estimates of new classes of q-starlike and q-convex functions of complex order,” Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 135–145, 2016. View at: Publisher Site | Google Scholar | MathSciNet
17. M. I. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics. Second Series, vol. 37, no. 2, pp. 374–408, 1936. View at: Publisher Site | Google Scholar | MathSciNet
18. W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis (Tianjin '92), pp. 157–169, Internat. Press, Cambridge, MA, USA. View at: Google Scholar | MathSciNet
19. S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975. View at: Publisher Site | Google Scholar | MathSciNet

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