New subclasses of analytic and bi-univalent functions endowed with coefficient estimate problems

A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z < 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

Motivated by q-calculus, we define a new family of Σ, which is the family of bi-univalent analytic functions in the open unit disc U that is related to the Einstein function E(z). We establish estimates for the first two Taylor–Maclaurin coefficients |a2|, |a3|, and the Fekete–Szegö inequality a3−μa22 for the functions that belong to these families.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients | a n | of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

Certain subclasses of analytic and bi-univalent functions

In this research, a new subclass of analytic and bi-univalent function is defined on the open unit disc \(\mathbb{U}\) = {z \(\in\) \(\mathbb{C}\) : |z| < 1} using the integral operator and then the initial coefficient bounds for |a2| and |a3| are derived. Later on, the many interesting results are discussed which are obtained as the special cases of our main result.

The objective of this paper is to introduce and investigate two families of analytical and bi-univalent functions, and , with respect to symmetric conjugate points that are defined in the open unit disk and connected to a series of beta-negative binomial distributions. For functions in each of these families, we look into upper bounds for the initial Taylor-Maclaurin coefficients and .

In this paper, estimates for second and third MacLaurin coefficients of a new subclass of analytic and bi-univalent functions in the open unit disk are determined, and certain special cases are also indicated

In this work, we introduce a new subclas of bi-univalent functions which is defined by Hadamard product andsubordination in the open unit disk. and find upper bounds for the second and third coefficients for functions in this new subclass. Further, we generalize and improve some of the previously published results.

This paper introduces and explores a novel subclass $\mathcal{R}_{\Sigma}^{h,p}(\lambda,\gamma)$ of bi-univalent functions within the open unit disk $\mathbb{U}$. We establish upper bounds for the second, third, and fourth coefficients of functions belonging to this class. Furthermore, we derive estimates for the Fekete-Szegö problem in this context. The results, presented in this study extend and enhance several recent works by earlier authors.

‎In this paper‎, ‎we introduce and investigate a subclass‎ of analytic and bi-univalent functions which both $f(z)$ and $f^{-1}(z)$ are m-fold symmetric in the open unit disk U‎. Furthermore‎, ‎we find upper bounds for the initial coefficients $|a_{m‎ + ‎1}|$ and $|a_{2m‎ + ‎1}|$ for functions in this subclass‎. ‎The results presented in this paper would generalize and improve some recent works‎.

In the open unit disk, we introduce a new subclass of analytic and bi-univalent functions described by the Neutrosophic Poisson distribution series. The estimates for the Fekete--Szeg\"{o} type and the Taylor coefficients. Then, using the Horadam polynomials, the inequalities corresponding to the functions that belong to this recently discovered subclass are investigated. Along with some corollaries, we also go over the implications of the findings presented in this essay. Numerous branches of mathematics, science, and technology are anticipated to heavily rely on the neutrophilic Poisson distribution.

In this paper, we investigate a subclass Σ (γ, τ) of analytic and bi-univalent functions in the open unit disk u. For functions belonging to this subclass, we obtain an upper bound for the second Hankel determinant H2(2).

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to H", the linear invariant family of locally univalent functions of finite order or.If f(z) is subordinate to F(z) in the open unit disc, /'(0) > 0, and F(z) is in U" 1.65 < a < oo, then f'(z) is majorized by F'(z) in \z\ < (a + 1) -(a2 + 2a)vi.The result is sharp.I. Introduction.Let <S denote the set of all normalized analytic univalent functions in the open unit disc D. Let/(z), F(z) and <p(z) be analytic in |z| < r.We say that/(z) is majorized byLet tla be the set of all locally univalent (f'(z) = 0) analytic functions in D with order < a which are of the form f(z) = z A-.The family U" is known as the universal linear invariant family of order a [4].A concise summary and introduction to properties of linear invariant families which relate to the following material is contained in [1].The present paper concludes the proof of results announced in [1].Majorization-subordination theory begins with Biernacki who showed in 1936 that if/(z) is subordinate in D to F(z) (F(z) G <S), then/(z) is majorized by F(z) in |z| < 1/4.In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems but always under the stipulation that the dominant function F(z) is in <S (for greater detail see [1]).In 1951 Goluzin showed that if f(z) is majorized by a univalent function F(z), then f'(z) would be majorized by F'(z) in |z| < 0.12.He conjectured that majorization would always occur for jz| < 3 -y and this was proved by Tao Shah in 1958.In this paper we show that the result is actually true for functions in 11. and obtain the sharp radius of majorization as a + 1 -(a2 + 2a)^2 for 1.65 < o < oo.This yields 3 -y for the case a = 2.Our investigation shows that the important datum for majorization-subordination theory is not univalence, but the order of a linear invariant family.In particular, many classically derived estimates for univalent functions are true for functions of infinite valence.

n this paper, we introduce and investigate a new subclass of Bi-univalent functions defined in the open unit disk, associated with Chebyshev polynomials by applying $q$-derivative operator. Furthermore, We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class and also we obtain an estimation for Fekete-Szeg$\ddot{o}$ problem for this class.