Abstract
In this work, an explicit formula for a class of Bi-Bazilevic univalent functions involving differential operator is given, as well as the determination of upper bounds for the general Taylor-Maclaurin coefficient of a functions belong to this class, are established Faber polynomials are used as a coordinated system to study the geometry of the manifold of coefficients for these functions. Also determining bounds for the first two coefficients of such functions.
 In certain cases, our initial estimates improve some of the coefficient bounds and link them to earlier thoughtful results that are published earlier.
Highlights
IntroductionLet A be the class of all functions f(z) as the following form f(z) = z + ∑∞n=2 anzn , (z ∈ U)
Let A be the class of all functions f(z) as the following form f(z) = z + ∑∞n=2 anzn, (z ∈ U) (1.1)which are analytic and normalized in the open unit disk U = {z ∈ C ∶ |z| < 1}.let S be the subclass of A consist of all functions that are univalent functions in U
M. (7) to a general class of meromorphic Biunivalent functions, we use the instrument of the well-known Faber polynomial expansions to determine estimates for a general subclass of analytic Bi-Bazilevic functions
Summary
Let A be the class of all functions f(z) as the following form f(z) = z + ∑∞n=2 anzn , (z ∈ U). A function f ∈ S has an inverse f−1 is defined as follows f−1(f(z)) = z, (z ∈ U). If g = f−1 is the inverse of the function f ∈ S, g has a Maclaurin series expansion in some disk about the origin which is given by g(w) = f−1 = w − a2w2 + (2a22 − a3)w3. A function f ∈ A is called Bi-Bazilevic if both f and its inverse f−1 are Bazilevic in the unite disk U. For a function f(z)α given by (1.5), define the differential operator Γβm,μ,λ: Aα ⟶ Aα as follows: Γβ0,,μλf(z)α = f(z)α.
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