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Article

New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials

1
Vocational School of Social Sciences, Bingöl University, Bingöl 12000, Türkiye
2
Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum 25050, Türkiye
3
Department of Mathematics, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(11), 652; https://doi.org/10.3390/axioms11110652
Submission received: 11 October 2022 / Revised: 11 November 2022 / Accepted: 15 November 2022 / Published: 17 November 2022

Abstract

:
In this paper, we introduce and investigate new subclasses of bi-univalent functions with respect to the symmetric points in U = z C : z < 1 defined by Bernoulli polynomials. We obtain upper bounds for Taylor–Maclaurin coefficients a 2 , a 3 and Fekete–Szegö inequalities a 3 μ a 2 2 for these new subclasses.

1. Introduction

Let the class of analytic functions in U = z C : z < 1 , denoted by A , contain all the functions of the type
l z = z + k = 2 a k z k , ( z U ) ,
which satisfy the usual normalization condition l ( 0 ) = l ( 0 ) 1 = 0 .
Let S be the subclass of A consisting of all functions l A , which are also univalent in U. The Koebe one quarter theorem [1] ensures that the image of U under every univalent function l A contains a disk of radius 1 4 . Thus, every univalent function l has an inverse l 1 satisfying
l 1 l z = z , ( z U ) and l l 1 ω = ω , ( ω < r 0 ( l ) , r 0 ( l ) 1 4 ) .
If l and l 1 are univalent in U , then l A is said to be bi-univalent in U , and the class of bi-univalent functions defined in the unit disk U is denoted by Σ . Since l Σ has the Maclaurin series given by (1), a computation shows that m = l 1 has the expansion
m ω = l 1 ω = ω a 2 ω 2 + 2 a 2 2 a 3 ω 3 + .
The expression Σ is a non-empty class of functions, as it contains at least the functions
l 1 z = z 1 z , l 2 z = 1 2 log 1 + z 1 z ,
with their corresponding inverses
l 1 1 ω = ω 1 + ω , l 2 1 ω = e 2 ω 1 e 2 ω + 1 .
In addition, the Koebe function l ( z ) = z ( 1 z ) 2 Σ .
The study of analytical and bi-univalent functions is reintroduced in the publication of [2] and is then followed by work such as [3,4,5,6,7,8]. The initial coefficient constraints have been determined by several authors who have also presented new subclasses of bi-univalent functions (see [2,3,4,6,9,10,11]).
Consider α and β to be analytic functions in U . We say that α is subordinate to β , if a Schwarz function w exists that is analytic in U with w ( 0 ) = 0 and w ( z ) < 1 , z U such that
α ( z ) = β w ( z ) , z U .
This subordination is denoted by α β or α ( z ) β ( z ) , z U . Given that β is a univalent function in U, then
α ( z ) β ( z ) α ( 0 ) = β ( 0 ) and α ( U ) β ( U ) .
Using Loewner’s technique, the Fekete–Szegö problem for the coefficients of l S in [6] is
a 3 μ a 2 2 1 + 2 exp 2 μ 1 μ for 0 μ < 1 .
The elementary inequality a 3 a 2 2 1 is obtained as μ 1 . The coefficient functional
F μ ( l ) = a 3 μ a 2 2
on the normalized analytic functions l in the open unit disk U also has a significant impact on geometric function theory. The Fekete–Szegö problem is known as the maximization problem for functional F μ ( l ) .
Researchers were concerned about several classes of univalent functions (see [12,13,14,15]) due to the Fekete–Szegö problem, proposed in 1933 ([16]); therefore, it stands to reason that similar inequalities were also discovered for bi-univalent functions, and fairly recent publications can be cited to back up the claim that the subject still yields intriguing findings [17,18,19].
Because of their importance in probability theory, mathematical statistics, mathematical physics, and engineering, orthogonal polynomials have been the subject of substantial research in recent years from a variety of angles. The classical orthogonal polynomials are the orthogonal polynomials that are most commonly used in applications (Hermite polynomials, Laguerre polynomials, Jacobi polynomials, and Bernoulli). We point out [17,18,20,21,22,23,24] as more recent examples of the relationship between geometric function theory and classical orthogonal polynomials.
Fractional calculus, a classical branch of mathematical analysis whose foundations were laid by Liouville in an 1832 paper and is currently a very active research field [25], is one of many special functions that are studied. This branch of mathematics is known as the Bernoulli polynomials, named after Jacob Bernoulli (1654–1705). A novel approximation method based on orthonormal Bernoulli’s polynomials has been developed to solve fractional order differential equations of the Lane–Emden type [26], whereas in [27,28,29], Bernoulli polynomials are utilized to numerically resolve Fredholm fractional integro-differential equations with right-sided Caputo derivatives.
The Bernoulli polynomials B n ( x ) are often defined (see, e.g., [30]) using the generating function:
F x , t = t e x t e t 1 = n = 0 B n ( x ) n ! t n , t < 2 π ,
where B n x are polynomials in x, for each nonnegative integer n.
The Bernoulli polynomials are easily computed by recursion since
j = 0 n 1 j n B j x = n x n 1 , n = 2 , 3 , .
The initial few polynomials of Bernoulli are
B 0 x = 1 , B 1 x = x 1 2 , B 2 x = x 2 x + 1 6 , B 3 x = x 3 3 2 x 2 + 1 2 x , .
Sakaguchi [31] introduced the class S s * of functions starlike with respect to symmetric points, which consists of functions l S satisfying the condition
R e z l ( z ) l z l ( z ) > 0 , z U .
In addition, Wang et al. [32] introduced the class C s of functions convex with respect to symmetric points, which consists of functions l S satisfying the condition
R e z l z l z l ( z ) > 0 , z U .
In this paper, we consider two subclasses of Σ : the class S s Σ x of functions bi-starlike with respect to the symmetric points and the relative class C s Σ x of functions bi-convex with respect to the symmetric points associated with Bernoulli polynomials. The definitions are as follows:
Definition 1. 
l S s Σ x , if the next subordinations hold:
2 z l ( z ) l ( z ) l ( z ) F x , z ,
and
2 ω m ( ω ) m ω m ( ω ) F ( x , ω ) ,
where z , ω U , F x , z is given by (3), and m = l 1 is given by (2).
Definition 2. 
l C s Σ x , if the following subordinations hold:
2 z l z l z l ( z ) F x , z ,
and
2 ω m ( ω ) m ω m ( ω ) F ( x , ω ) ,
where z , ω U , F x , z is given by (3), and m = l 1 is given by (2).
Lemma 1 
([33], p. 172). Suppose that c ( z ) = n = 1 c n z n , c ( z ) < 1 , z U , is an analytic function in U . Then,
c 1 1 , c n 1 c 1 2 , n = 2 , 3 , .

2. Coefficients Estimates for the Class S s Σ x

We obtain upper bounds of a 2 and a 3 for the functions belonging to the class S s Σ x .
Theorem 1. 
If l S s Σ x , then
a 2 B 1 x 6 B 1 x ,
and
a 3 B 1 x 2 + B 1 x 2 4 .
Proof. 
Let l S s Σ x and m = l 1 . From definition in (6) and (7), we have
2 l ( z ) z l z l ( z ) = F ( x , φ ( z ) ) ,
and
2 ω m ( ω ) m ω m ( ω ) = F ( x , χ ω ) ,
where φ and χ are analytic functions in U given by
φ ( z ) = r 1 z + r 2 z 2 + ,
χ ω = s 1 ω + s 2 ω 2 + ,
and φ 0 = χ 0 = 0 , and φ ( z ) < 1 , χ ω < 1 , z , ω U .
As a result of Lemma 1,
r k 1 and s k 1 , k N .
If we replace (14) and (15) in (12) and (13), respectively, we obtain
2 z l ( z ) l z l ( z ) = B 0 x + B 1 x φ z + B 2 x 2 ! φ 2 z + ,
and
2 ω m ( ω ) m ω m ( ω ) = B 0 x + B 1 x χ ω + B 2 x 2 ! χ 2 ω + .
In view of (1) and (2), from (17) and (18), we obtain
1 + 2 a 2 z + 2 a 3 z 2 + = 1 + B 1 x r 1 z + B 1 x r 2 + B 2 x 2 ! r 1 2 z 2 +
and
1 2 a 2 ω + ( 4 a 2 2 2 a 3 ) ω 2 + = 1 + B 1 x s 1 ω + B 1 x s 2 + B 2 x 2 ! s 1 2 ω 2 + ,
which yields the following relations:
2 a 2 = B 1 x r 1 ,
2 a 3 = B 1 x r 2 + B 2 x 2 ! r 1 2 ,
and
2 a 2 = B 1 x s 1 ,
4 a 2 2 2 a 3 = B 1 x s 2 + B 2 x 2 ! s 1 2 .
From (19) and (21), it follows that
r 1 = s 1 ,
and
8 a 2 2 = B 1 x 2 r 1 2 + s 1 2
a 2 2 = B 1 x 2 r 1 2 + s 1 2 8 .
Adding (20) and (22), using (24), we obtain
a 2 2 = B 1 x 3 ( r 2 + s 2 ) 4 ( B 1 ( x ) 2 B 2 x ) .
Using relation (5), from (16) for r 2 and s 2 , we get (10).
Using (23) and (24), by subtracting (22) from relation (20), we get
a 3 = B 1 x r 2 s 2 + B 2 x 2 ! ( r 1 2 s 1 2 ) 4 + a 2 2 = B 1 x r 2 s 2 + B 2 x 2 ! ( r 1 2 s 1 2 ) 4 + B 1 x 2 r 1 2 + s 1 2 8 .
Once again applying (23) and using (5), for the coefficients r 1 , s 1 , r 2 , s 2 , we deduce (11). □

3. The Fekete–Szegö Problem for the Function Class S s Σ x

We obtain the Fekete–Szegö inequality for the class S s Σ x due to the result of Zaprawa; see [19].
Theorem 2. 
If l given by (1) is in the class S s Σ x where μ R , then we have
a 3 μ a 2 2 B 1 x 2 , i f h ( μ ) 1 4 , 2 B 1 x h ( μ ) , i f h ( μ ) 1 4 ,
where
h μ = 3 ( 1 μ ) B 1 x 2 .
Proof. 
If l S s Σ x is given by (1), from (25) and (26), we have
a 3 μ a 2 2 = B 1 x r 2 s 2 4 + ( 1 μ ) a 2 2 = B 1 x r 2 s 2 4 + ( 1 μ ) B 1 x 3 ( r 2 + s 2 ) 4 ( B 1 x 2 B 2 x ) = B 1 x r 2 4 s 2 4 + 1 μ B 1 x 2 r 2 4 ( B 1 x 2 B 2 x ) + 1 μ B 1 x 2 s 2 4 ( B 1 x 2 B 2 x ) = B 1 x h μ + 1 4 r 2 + h μ 1 4 s 2 ,
where
h μ = ( 1 μ ) B 1 x 2 4 ( B 1 x 2 B 2 x )
Now, by using (5)
a 3 μ a 2 2 = x 1 2 h μ + 1 4 r 2 + h μ 1 4 s 2 ,
where
h μ = 3 ( 1 μ ) x 1 2 2 .
Therefore, given (5) and (16), we conclude that the necessary inequality holds. □

4. Coefficients Estimates for the Class C s Σ x

We will obtain upper bounds of a 2 and a 3 for the functions belonging to a class C S Σ x .
Theorem 3. 
If l C s Σ x , then
a 2 B 1 x B 1 x 6 B 1 x 2 8 B 2 x ,
and
a 3 B 1 x 6 + B 1 x 2 16 .
Proof. 
Let l C s Σ x and m = l 1 . From (8) and (9), we get
2 z l z l z l ( z ) = F ( x , φ ( z ) ) ,
and
2 ω m ( ω ) m ω m ( ω ) = F ( x , χ ω )
where φ and χ are analytic functions in U given by
φ ( z ) = r 1 z + r 2 z 2 + ,
χ ω = s 1 ω + s 2 ω 2 + ,
where φ 0 = χ 0 = 0 , and φ ( z ) < 1 , χ ω < 1 , z , ω U .
As a result of Lemma 1,
r k 1 and s k 1 , k N .
If we replace (31) and (32) in (29) and (30), respectively, we obtain
2 z l z l z l ( z ) = B 0 x + B 1 x φ z + B 2 x 2 ! φ 2 z + ,
and
2 ω m ( ω ) m ω m ( ω ) = B 0 x + B 1 x χ ω + B 2 x 2 ! χ 2 ω + .
In view of (1) and (2), from (34) and (35), we obtain
1 + 4 a 2 z + 6 a 3 z 2 + = 1 + B 1 x r 1 z + B 1 x r 2 + B 2 x 2 ! r 1 2 z 2 +
and
1 4 a 2 ω + 12 a 2 2 6 a 3 ω 2 + = 1 + B 1 x s 1 ω + B 1 x s 2 + B 2 x 2 ! s 1 2 ω 2 + ,
which yields the following relations:
4 a 2 = B 1 x r 1 ,
6 a 3 = B 1 x r 2 + B 2 x 2 ! r 1 2 ,
and
4 a 2 = B 1 x s 1 ,
12 a 2 2 6 a 3 = B 1 x s 2 + B 2 x 2 ! s 1 2 .
From (36) and (38), it follows that
r 1 = s 1 ,
and
32 a 2 2 = B 1 x 2 r 1 2 + s 1 2
a 2 2 = B 1 x 2 r 1 2 + s 1 2 32 .
Adding (37) and (39), using (41), we obtain
a 2 2 = B 1 x 3 ( r 2 + s 2 ) 4 ( 3 B 1 x 2 4 B 2 x ) .
Using relation (5), from (33) for r 2 and s 2 , we get (27). Using (40) and (41), by subtracting (39) from relation (37), we get
a 3 = B 1 x r 2 s 2 + B 2 x 2 ! r 1 2 s 1 2 12 + a 2 2 = B 1 x r 2 s 2 + B 2 x 2 ! r 1 2 s 1 2 12 + B 1 x 2 r 1 2 + s 1 2 32 .
Once again applying (40) and using (5), for the coefficients r 1 , s 1 , r 2 , s 2 , we deduce (28). □

5. The Fekete–Szegö Problem for the Function Class C s Σ x

We obtain the Fekete–Szegö inequality for the class C s Σ x due to the result of Zaprawa; see [19].
Theorem 4. 
If l given by (1) is in the class C s Σ x where μ R , then, we have
a 3 μ a 2 2 B 1 x 6 , i f h ( μ ) 1 12 , 2 B 1 x h ( μ ) , i f h ( μ ) 1 12 ,
where
h μ = ( 1 μ ) B 1 x 2 4 ( 3 B 1 x 2 4 B 2 x ) .
Proof. 
If l C s Σ x is given by (1), from (42) and (43), we have
a 3 μ a 2 2 = B 1 x r 2 s 2 12 + ( 1 μ ) a 2 2 = B 1 x r 2 s 2 12 + ( 1 μ ) B 1 x 3 ( r 2 + s 2 ) 4 ( 3 B 1 x 2 4 B 2 x ) = B 1 x r 2 s 2 12 + 1 μ B 1 x 2 r 2 4 ( 3 B 1 x 2 4 B 2 x ) + 1 μ B 1 x 2 s 2 4 ( 3 B 1 x 2 4 B 2 x ) = B 1 x h μ + 1 12 r 2 + h μ 1 12 s 2 ,
where
h μ = ( 1 μ ) B 1 x 2 4 ( 3 B 1 x 2 4 B 2 x ) .
Now, by using (5)
a 3 μ a 2 2 = x 1 2 h μ + 1 12 r 2 + h μ 1 12 s 2 ,
where
h μ = ( 1 μ ) x 1 2 2 4 ( 3 x 1 2 2 4 ( x 2 x + 1 6 ) ) .
Therefore, given (5) and (33), we conclude that the required inequality holds. □

6. Conclusions

We introduce and investigate new subclasses of bi-univalent functions in U associated with Bernoulli polynomials and satisfying subordination conditions. Moreover, we obtain upper bounds for the initial Taylor–Maclaurin coeffcients a 2 , a 3 and Fekete–Szegö problem a 3 μ a 2 2 for functions in these subclasses.
The approach employed here has also been extended to generate new bi-univalent function subfamilies using the other special functions. The researchers may carry out the linked outcomes in practice.

Author Contributions

Conceptualization, M.B., M.Ç. and L.-I.C.; methodology, M.Ç. and L.-I.C.; software, M.B., M.Ç. and L.-I.C.; validation, M.Ç. and L.-I.C.; formal analysis, M.B., M.Ç. and L.-I.C.; investigation, M.B., M.Ç. and L.-I.C.; resources, M.B., M.Ç. and L.-I.C.; data curation, M.B., M.Ç. and L.-I.C.; writing—original draft preparation, M.B., M.Ç. and L.-I.C.; writing—review and editing, M.B., M.Ç. and L.-I.C.; visualization, M.B., M.Ç. and L.-I.C.; supervision, M.Ç. and L.-I.C.; project administration, M.Ç. and L.-I.C.; funding acquisition, L.-I.C. All authors have read and agreed to the published version of the manuscript.

Funding

The research was partially funded by the project 38PFE, which was part of the PDI-PFE-CDI-2021 program.

Data Availability Statement

Not applicable.

Acknowledgments

We thank the referees for their insightful suggestions and comments to improve this paper in its present form.

Conflicts of Interest

The authors declare no conflict of interest.

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Buyankara, M.; Çağlar, M.; Cotîrlă, L.-I. New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials. Axioms 2022, 11, 652. https://doi.org/10.3390/axioms11110652

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Buyankara M, Çağlar M, Cotîrlă L-I. New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials. Axioms. 2022; 11(11):652. https://doi.org/10.3390/axioms11110652

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Buyankara, Mucahit, Murat Çağlar, and Luminiţa-Ioana Cotîrlă. 2022. "New Subclasses of Bi-Univalent Functions with Respect to the Symmetric Points Defined by Bernoulli Polynomials" Axioms 11, no. 11: 652. https://doi.org/10.3390/axioms11110652

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