Abstract
Motivated by q-calculus, subordination principle, and the second Einstein function, we define two families of bi-univalent analytic functions on the open unit disc of the complex plane. We deduce estimates for the first two Maclaurin’s coefficients and the Fekete-Sezgö functional inequalities for the functions that belong to these families of functions.
1. Introduction and Basic Concepts
Let denotes the collection of all functions with the following series representation: which are analytic and univalent in the open unit disc and satisfy the usual normalization condition . Also, an important class of functions will be called , defines the family of functions with the restrictions that the image domain of ( is a convex function with in ) is symmetric along the real axis and starlike about with .
In 1980, Gradshteyn and Ryzhik [1] give an expression of the Bernoulli polynomials which have important applications in number theory and classical analysis. They appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials. They are also used for representing the remainder term of the composite Euler-Maclaurin quadrature rule.
The Bernoulli polynomials are usually defined (see, e.g., [2]) by means of the generating function: where are polynomials in , for each nonnegative integer .
The Bernoulli polynomials are easily computed by recursion since
The first few Bernoulli polynomials are
Furthermore, Bernoulli numbers are produced directly by putting in Bernoulli polynomials. The first few Bernoulli numbers are
Moreover, Bernoulli numbers can be generated by means of the so-called Einstein function :
In mathematics, the Einstein function is a name occasionally used for one of the functions (see [3–6]):
It is easily noticed that both and have these nice properties, but and are not (see Figure 1); the range of and ( and are convex functions) is symmetric along the real axis and starlike about and .
(a)
(b)
(c)
(d)
The series representation is given by (one can type “Maclaurin series for ” in [7]) where is the Bernoulli number.
El-Qadeem et al. [8] have introduced some results related to the first Einstein function . Here, we will deal with the second Einstein function . Note that (indeed ), i.e., . Thus, we shall define the function: where . It is obvious that is a convex domain, symmetric along the real axis and starlike about and ; moreover, . This proves that
Example 1. (i)If , then is not a convex function, see Figure 2(a)(ii)If , then s.t. , see Figure 2(b)(iii)If , then ; also, is a convex function, see Figure 2(c)
(a)
(b)
(c)
Now, let be the subfamily of consisting of all functions of the form (1) which are univalent in . It is well known, by using the Koebe one-quarter theorem [9], that every univalent function containing a disc of radius has an inverse function , which is defined by
A function is said to be bi-univalent in if both and are univalent in . Let denotes the subfamily of , consisting of all biunivalent functions defined on the unit disc . Since has the Maclaurin series expansion given by (1), a simple calculation shows that its inverse has the series expansion:
Examples of functions in the class are and so on. However, the familiar Koebe function is not a member of . Other common examples of functions in such as are also not members of .
Now, we introduce some notes about the q-difference operator which uses in investigating our main families. In view of Annaby and Mansour [10], the q-difference operator is defined by
Thus, for the function denoted by (1), we have where
Definition 1 (see [11, 12]). An analytic function is said to be subordinate to another analytic function , written as , if there exists a Schwarz function , which is analytic in with and , such that . In particular, if the function is univalent in , then we have the following equivalence: Our aim in this article is to introduce two families of analytic bi-univalent function related to the modified Einstein function . Furthermore, we get estimations to , , and also the Fekete Sezgö inequalities for the functions that belong to these two families.
Definition 2. Consider , , , and . The function is said to be in if it is satisfying where is given by (11) and
Definition 3. Consider , , , and . The function is said to be in if it is satisfying where is given by (11) and .
Remark 4. Specializing the parameters , , , , and in Definitions 2 and 3, then we have the following subfamilies: (i)If and , then is the subfamily of , consisting of functions which satisfy the following conditions:where is given by (11), is defined by (9), and (ii)If , , and , then let be the family of bi-univalent convex functions with respect to Einstein function, consisting of the functions which satisfy the following conditions:where is given by (11), is defined by (9), and (iii)If and , then let be the family of bi-univalent starlike functions with respect to Einstein function, consisting of functions which satisfy the following conditions:where is given by (11), is defined by (9), and (iv)If and , then is the subfamily of , consisting of functions which satisfy the following conditions:where is given by (11), is defined by (9), and (v)If , , and , then is the subfamily of , consisting of functions which satisfy the following conditions:where is given by (11), is defined by (9), and
Lemma 5 (see [13, 14]). Let and . If , then
Lemma 6 (see [9]). Suppose that is analytic in the unit open disc with , , and that then
2. Main Results
Unless otherwise mentioned, we assume in the remainder of this article that , , , , , , and .
Theorem 7. Let , then where
Proof. Let and be in , then, they satisfy (18) and (19), respectively. But according to Definition 1 and Lemma 6, there exist two functions and of the form
such that
After some simple calculations, we deduce
Also,
where are stated in (30).
By substituting from (34), (35), (36), and (37) into (32) and (33) with comparing the coefficient in both sides, we obtain
As a direct result of equations (38) and (40), we get
and also,
By adding (39) to (41), then using (43), we obtain
Equations (42) and (44) together with using Lemma 6 implies that
But from equation (38), we can deduce
By using (46) into (45), we obtain
Further, from (39) and (41) with using (42), we get
Thus, by virtue of Lemma 6, we find
Since for all . Then, we conclude
Thus, the proof is completed.
Theorem 8. Let , then where
Proof. Suppose and be in , then it satisfies (18) and (19). According to Definition 1 and Lemma 6, there exist two functions of the form
such that
With some simple calculations, we get
By substituting from (34), (35), (57), and (58) into (55) and (56) with comparing the coefficient in both sides, we conclude
From (59) and (61), we obtain
and also,
By adding (60) to (62) with using (64), we get
In view of Lemma 6, equation (65) together with (63) implies that
On the other hand, from equation (59), we can write
By using (67) into (66), we get
where is defined in (52).
Further, by subtracting (62) from (60) and using (63), we have
In view of Lemma 6, equation (69) together with (67) implies that
By the virtue of (68), we can get the desired result. Thus, we complete the proof.
Theorem 9. Suppose and , then where and , and are given by (30).
Proof. To investigate the desired result, we first subtract (41) from (39) with using (42), we get
Thus,
As a result of subsequent computations performed by using (44), we obtain
where given by (72).
But in view of Kanas et al. [15] and (28), we can obtain
Now, applying Lemma 5 to (75), we can obtain the desired result directly. Thus, we complete the proof.
Theorem 10. Let us consider and , then where is defined by (52).
Proof. In order to investigate the desired result (77), we first subtract (62) from (60) with taking in consideration (63), we conclude By virtue of (65), we can get that By applying Lemma 5 to (79) with using (76), we obtain the required result which completes the proof.
3. Set of Corollaries
In this part, we introduce some corollaries by specializing the values of the given parameters and , and taking the limit when , in our main results.
Put and in Theorems 7, 9, then we have the following corollary:
Corollary 11. Let belong to , then
Put and in Theorems 7, 9, then we get the following corollary:
Corollary 12. Let belong to , then
Put and in Theorems 7, 9, then we obtain the following:
Corollary 13. Let belong to , then
Put and in Theorems 8, 10, then we have the following corollary:
Corollary 14. Let belong to , then
Put and in Theorems 8, 10, then we have the following corollary:
Corollary 15. Let belong to , then
Data Availability
No data have been used.
Conflicts of Interest
The authors confirm no competing interests.
Authors’ Contributions
The authors read and approved the final manuscript.