Research Article | Open Access

# Maclaurin Coefficient Estimates for New Subclasses of Bi-univalent Functions Connected with a -Analogue of Bessel Function

**Academic Editor:**Wei-Shih Du

#### Abstract

In this paper, we introduce new subclasses of the function class of bi-univalent functions connected with a -analogue of Bessel function and defined in the open unit disc. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients and for functions in these new subclasses.

#### 1. Introduction, Definitions, and Preliminaries

The theory of -calculus plays an important role in many areas of mathematical, physical, and engineering sciences. Jackson (see [1, 2]) was the first to have some applications of the -calculus and introduced the -analogue of the classical derivative and integral operators (see also [3]). Let denote the class of analytic functions of the form and be the subclass of which are univalent functions in .

If is given by
then, the *Hadamard* (*or convolution*) *product* of and is defined by

If and are analytic functions in , we say that *is subordinate to*, written , if there exists a *Schwarz function*, which is analytic in , with , and for all , such that , . Furthermore, if the function is univalent in , then we have the following equivalence (see [4, 5]):

The *Bessel function of the first kind of order* is defined by the infinite series (see [6])
where stands for the *Gamma function*. Recently, Szász and Kupán [7] investigated the univalence of the normalized Bessel function of the first kind defined by (see also [8–10])
For , the *-derivative operator* for is defined by
where

Using the definition formula (8), we will define the next two products:
(i)For any nonnegative integer , the *-shifted factorial* is given by
(ii)For any positive number , the *-generalized Pochhammer symbol* is defined by

For , , and , El-Deeb and Bulboacă [11] define the function by

A simple computation shows that where the function is given by

Using the definition of -derivative along with the idea of convolutions, El-Deeb and Bulboacă [11] introduce the linear operator defined by where

*Remark 1. *From the definition relation (14), we can easily verify that the next relations hold for all :

The Koebe one quarter theorem (see [12]) proves that the image of under every univalent function contains a disk of radius Therefore, every function has an inverse satisfying where

A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (1). For a brief history and interesting examples in the class , see [13]. Brannan and Taha [14] (see also [15–17]) introduced certain subclasses of the bi-univalent functions class similar to the familiar subclasses and of starlike and convex functions of order , respectively (see [13]). Thus, following Brannan and Taha [14], a function is said to be in the class of strongly bi-starlike functions of order if each of the following conditions is satisfied: where the function is given by and is the extension of to The classes and of bi-starlike functions of order and biconvex functions of order corresponding to the function classes and were also introduced analogously. For each of the function classes and they found nonsharp estimates on the first two Taylor-Maclaurin coefficients and (for details, see [14, 17]).

The objective of the present paper is to introduce new subclasses of the function class and find estimates on the coefficients and for functions in these new subclasses of the function class employing the techniques used earlier by Srivastava et al. [18].

Now, we define the subclasses of functions , and as follows:

*Definition 2. *Let be given by (1), then is said to be in the class if the following conditions are satisfied:
where the function is given by (2).

Putting we obtain that , where

*Definition 3. *Let be given by (1), then is said to be in the class if the following conditions are satisfied:
where the function is given by (2).

Putting we obtain that , where

To prove our results, we need the following lemma.

Lemma 4 (see [19], Lemma 3). *If then for each , where is the family of all functions analytic in for which for *

*Definition 5. *Let be given by (1), and in have the forms
then is said to be in the class if the following conditions are satisfied:
where the function is given by (2).

#### 2. Coefficient Bounds for the Function Class

Unless otherwise mentioned, we assume throughout this paper that

Theorem 6. *Let be given by (1) which belongs to the class , then
where , , are given by (15).*

*Proof. *It follows from (21) and (22) that
where and in and have the forms
Now, equating the coefficients in (34) and (35), we get
From (38) and (40), we get
From (39), (41), and (43), we obtain
Applying Lemma 4 for the coefficients and , we immediately have
This gives the bound on as asserted in (32).

Next, in order to find the bound on , by subtracting (41) from (39), we get
It follows from (42), (43), and (46) that
Applying Lemma 4 once again for the coefficients , and , we immediately have
This completes the proof of Theorem 6.

#### 3. Coefficient Bounds for the Function Class

Theorem 7. *Let be given by (1) which belongs to the class , then
where , , are given by (15).*

*Proof. *It follows from (24) and (25) that
where and have the forms (36) and (37), respectively. Equating the coefficients in (51) and (52), we get
From (53) and (55), we get
From (54) and (56), we obtain
Applying Lemma 4 for the coefficients and , we immediately have
This gives the bound on as asserted in (49).

Next, in order to find the bound on , by subtracting (56) from (54), we get
It follows from (58) and (61) that
Applying Lemma 4 once again for the coefficients and , we immediately have

This completes the proof of Theorem 6.

#### 4. General Coefficient Bounds for the Function Class

This section begins by finding the estimates on the coefficients and for functions in the class

Theorem 8. *Let be given by (1) which belongs to the class , then
where , , are given by (15).*

*Proof. *It follows from (29) and (30) that
where and are given by (27) and (28). Now, equating the coefficients of and in (66) and (67), we get
From (68) and (70), we get
From (69) and (71), we obtain
Therefore, we obtain from the equations (73) and (74) that
respectively, this gives the bound on as asserted in (64).

Next, in order to find the bound on , by subtracting (71) from (69), we get
It follows from (73) into (76) that
We immediately have
On the other hand, upon substituting the value of from (74) into (76), we get
Consequently, we get

This completes the proof of Theorem 8.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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#### Copyright

Copyright © 2020 Sheza M. El-Deeb. 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.