On Fekete–Szegö Problems for Certain Subclasses of Analytic Functions Defined by Differential Operator Involving -Ruscheweyh Operator

Abstract

In this paper, we define a new derivative operator involving -Ruscheweyh differential operator using convolution. Using this new operator, we introduce two new classes of analytic functions and obtain the Fekete–Szegö inequalities.

1. Introduction

The applications of -calculus are important and pivotal as they contributed worthy of noticed expansion in geometric function theory. In 1908, Jackson, was the first mathematician to develop the application of -calculus in a systematic way [1, 2]. Then, Aral and Gupta [3] proposed -analogue of Baskakov and Durrmeyer operator depending on -calculus; Mohammed and Darus [4] defined a new operator involving the -hypergeometric function. Some other applications of -calculus are studied by the authors [5, 6] and Elhaddad et al. [7]. Recently, many mathematicians have worked intensively in this field (see [8–12]) and obtained various results.

Let be the class of functions of the form

which are analytic in and we denote by the subclass of that are consisted of one-to-one (univalent) functions in .

The convolution of functions as in (1.1) and will be:

Let and are analytic functions in , then we say that is subordinate to denoted by in , if there exists a Schwarz function which is analytic in with and () such that .

Now, we give some notations and definitions of the principal terms of -calculus by assuming , as follows:

(1) The -number is defined by:

(2) The -factorial is defined by:

(3) The -derivative of a function is defined by:

where will be used throughout the article.

As given by (1), then

The authors in [6] introduced a -differential operator by:

where , , .

In 2014, Aldweby and Darus in [9] introduced -Ruscheweyh operator by:

Using convolution on , we define new differential operator by:

where

Remarks. (i)When , , , , and , we get -Sãlãgean differential operator introduced in [13].(ii)When , , , , , and , we get Sãlãgean differential operator introduced in [14].(iii)When , , , , , and , then we get Al-Oboudi differential operator introduced in [15].(iv)When and , we get Ramadan and Darus operator introduced in [16].(v)When , we get Alsoboh and Darus operator introduced in [6].(vi)When , then we get -Ruscheweyh operator introduced in [9].(vii)When and , then we have Ruscheweyh operator introduced in [17].(viii)When , , , , and , we get [18].

Many sublasses of analytic functions have been introduced by many different authors, for example, Ma and Minda [19], Ravichandran et al. [20], Seoudy and Aouf [12] and others. These works have inspired our introduction of the new subclasses and of , involving the differential operator and the principle of subordination as:

Definition 1. Let be the subclass of functions which are analytic and univalent in and for which is convex with and for . A function is said to be in the class if it satisfies the subordination condition:

Definition 2. A function is said to be in the class if it satisfies the subordination condition:

Note that:(i)When and , then and [12].(ii)When , , and , then and [20].(iii)When , , and , then , and [19].

2. Main Results

In this section, we obtain the Fekete-Szegö inequalities for the subclasses and , by assuming , , , and . In order to prove our results, we use the following lemmas of Ma and Minda [19].

Lemma 3. If is a function with positive real part in the open unit disk and , then

Lemma 4. If is a function with positive real part in and , then

To get our results, we use the similar methods studied by Alsoboh and Darus [5], Elhaddad and Darus [11], and Seoudy and Aouf [12].

Theorem 5. Let with , and given by (1) belongs to , then

Proof. Let , then there is a Schwarz function which is analytic in with , and in , such thatNow, we define the function with and by:Since is a Schwarz function, therefore,Now, substitute (18) in (16), we getFrom Equation (19), we getandTherefore,whereby applying Lemma 3, we get our result. This completes the proof.

Next, we prove the following theorem for the subclass .

Theorem 6. Let with , and given by (1) belongs to , then

Proof. Let , then there is a Schwarz function , analytic in with , and in such thatFrom the equations (17), (18), and last equality, we haveFrom Equation (26), we can findandTherefore,whereby applying Lemma 3, we get our result and this completes the proof.

When in previous two theorems, we obtain the result for functions in the classes and , respectively, which improved by Seoudy and Aouf [12]:

Corollary 7. Let with , and given by (1) belong to , thenThe result is sharp.

Corollary 8. Let with , and given by (1) belong to , then

The result is sharp.

For and taking in Theorem 5, we obtain the following corollary.

Corollary 9. (see [20]). Let with , and given by (1)belong to , thenThe result is sharp.

Next, by using Lemma 4, we obtain the following theorems:

Theorem 10. Let with , and given by (1) belong to , then where

Proof. Applying Lemma 4 to the Equations (22) and (23), we have three cases:
Case (1): If Therefore,Case (2): If Case (3): If This completes our proof.

Theorem 11. Let with , and given by (1) belong to , thenwhere

Proof. Applying Lemma 4 to the Equations (29) and (30), then we have
Case (1): If Case (2): If Case (3): If This completes the proof.

When in Theorems 5 and 6, we obtain the Fekete-Szegö inequalities for the subclasses and , respectively [12].

Corollary 12. Let with , and given by (1) belong to , with thenwhereThe result is sharp.

Corollary 13. Let with , and given by (1) belong to , with thenwhereThe result is sharp.

3. Conclusion

Fekete-Szegö problems have always been the main interests of many researchers in geometric function theory. Many studies related to Fekete-Szegö revolved around classes of analytic normalised univalent functions. In this particular work, the Fekete-Szegö inequality is obtained for functions in more general classes denoted by and , respectively, using a new differential operator associated with -calculus.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Funding

The work here is supported by UKM grant: GUP-2019-032.

Acknowledgments

We would like to thank the referee for the suggestions given to improve the content of the article.