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Article

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

by
1,
Murat Çağlar
2 and
Luminiţa-Ioana Cotîrlă
3,*
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.
MSC:
30C45; 30C50

## 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.

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

AMA Style

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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