Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination

Abstract

In this paper, we introduce new subclasses RΣ,b,cμ,αλ,δ,τ,Φ and KΣ,b,cμ,αλ,δ,η,Φ of bi-univalent functions in the open unit disk U by using quasi-subordination conditions and determine estimates of the coefficients a2 and a3 for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses RΣ,b,cμ,αλ,δ,τ,Φ and KΣ,b,cμ,αλ,δ,η,Φ, as there is a symmetry for the estimations of the coefficients a2 and a3 for all the subclasses defind in our this paper.

Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions

“…Let Σ denote the class of all functions f ∈ A that are bi-univalent in U given by (7). For interesting subclasses of functions in the class Σ, see [14][15][16][17][18][19]. Now, let us define the linear operator…”

Bi-Univalent Problems Involving Certain New Subclasses of Generalized Multiplier Transform on Analytic Functions Associated with Modified Sigmoid Function

“…To obtain the bound on a 3 as contained in (19), subtract (24) from ( 22) and then use (29) in the result of the difference, then…”

“…Extensions, generalizations, and improvements of starlikeness criteria for certain subclasses of analytic and bi-univalent functions were obtained considering coefficient estimates [17] and certain coefficient estimates were also given for particular families of bi-Bazilevic functions of the Ma-Minda type involving the Hohlov operator [18]. Quasi-Subordination was used for investigating new subclasses for bi-univalent functions in [19] and an integral operator based upon Lucas polynomial was also involved in the study of subclasses of bi-univalent functions [20]. New subclasses were introduced and studied considering aspects regarding argument and real part of certain linear combinations involving m-fold symmetric bi-univalent functions in [21].…”

On Some Sandwich Results of Univalent Functions Related by Differential Operator

“…Using the results (see [11,[18][19][20][21][22][23][24][25][26][27][28][29][30][31]) t𝑜 o𝑏𝑡ain suffici𝑒nt co𝑛𝑑itions for normalized an𝑎lytic functi𝑜ns to s𝑎𝑡isfy:…”