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

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 byThese 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.