Journal of Function Spaces
Volume 2017 (2017), Article ID 7156738, 5 pages
https://doi.org/10.1155/2017/7156738

Research Article

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

Huda Aldweby¹ and Maslina Darus²

¹Department of Mathematics, Faculty of Science, AL Asmarya Islamic University, Zliten, Libya
²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, Malaysia

Correspondence should be addressed to Maslina Darus; maslina@ukm.edu.my

Received 15 April 2017; Accepted 2 August 2017; Published 7 September 2017

Academic Editor: Henryk Hudzik

Copyright © 2017 Huda Aldweby and Maslina Darus. 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.

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 [3], Aral et al. [4], Li et al. [5], and many others (see [6–15]).

Making use of the -derivative, we define the subclasses and of the class for by

These classes are also studied and introduced by Seoudy and Aouf [16].

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 [8] that will be used throughout.

Definition 1 (see [8]). 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 [19].

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 [18]). If of positive real part is in and is a complex number, then

The result is sharp given by

Lemma 5 (see [18]). 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 [16].

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.

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