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

: 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 (cid:12)(cid:12) a 3 − µ a 22 (cid:12)(cid:12) for these new subclasses.


Introduction
Let the class of analytic functions in U = {z ∈ C : |z| < 1}, denoted by A, contain all the functions of the type 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 The expression Σ is a non-empty class of functions, as it contains at least the functions with their corresponding inverses 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 This subordination is denoted by Using Loewner's technique, the Fekete-Szegö problem for the coefficients of l ∈ S in [6] is The elementary inequality a 3 − a 2 2 ≤ 1 is obtained as µ → 1.The coefficient functional 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 integrodifferential equations with right-sided Caputo derivatives.
The Bernoulli polynomials B n (x) are often defined (see, e.g., [30]) using the generating function: where B n (x) are polynomials in x, for each nonnegative integer n.
The Bernoulli polynomials are easily computed by recursion since The initial few polynomials of Bernoulli are Sakaguchi [31] introduced the class S * s of functions starlike with respect to symmetric points, which consists of functions l ∈ S satisfying the condition 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 In this paper, we consider two subclasses of Σ: the class S Σ s (x) of functions bi-starlike with respect to the symmetric points and the relative class C Σ s (x) of functions bi-convex with respect to the symmetric points associated with Bernoulli polynomials.The definitions are as follows: where z, ω ∈ U, F(x, z) is given by (3), and m = l −1 is given by (2).

The Fekete-Szegö Problem for the Function Class S Σ s (x)
We obtain the Fekete-Szegö inequality for the class S Σ s (x) due to the result of Zaprawa; see [19].
Theorem 2. If l given by ( 1) is in the class S Σ s (x) where µ ∈ R, then we have Proof.If l ∈ S Σ s (x) is given by (1), from ( 25) and ( 26), we have where Now, by using ( 5) where Therefore, given ( 5) and ( 16), we conclude that the necessary inequality holds.

The Fekete-Szegö Problem for the Function Class C Σ s (x)
We obtain the Fekete-Szegö inequality for the class C Σ s (x) due to the result of Zaprawa; see [19].