Abstract
In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions defined in the unit disk O and establish upper bounds for the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szeg¨o problem for thesefamilies. Further, we point out several certain special cases for our results.
Highlights
Indicate by A the collection of functions U that are holomorphic in the unit disk O = {ξ ∈ C : |ξ| < 1} that have the shape:
A function U ∈ A is said to be bi-univalent in O if both U and U−1 are univalent in O, let we name by the notation E the set of bi-univalent functions in O satisfying (1.1)
Assume that δ ≥ 0, 0 ≤ λ ≤ 1 and 0 ≤ θ < 1, a function U ∈ E is called in the family WN E(δ, λ, θ; x) if it fulfills the subordinations: (1 − δ) (1 − λ) ξ (U ∗ Iθ) (ξ) + λ 1 + ξ (U ∗ Iθ) (ξ) + δ λξ2 (U ∗ Iθ) (ξ) + ξ (U ∗ Iθ) (ξ)
Summary
Let S stands for the subfamily of the collection A consisting of functions in O satisfying (1.1) that are univalent in O. A function U ∈ A is said to be bi-univalent in O if both U and U−1 are univalent in O, let we name by the notation E the set of bi-univalent functions in O satisfying (1.1). For functions U ∈ E is still not completely addressed for many of the subfamilies of the bi-univalent function class E. The Fekete-Szegö functional a3 − μa for U ∈ S is well known for its rich history in the field of Geometric Function Theory. Its origin was in the disproof by Fekete and Szegö [13] of the Littlewood-Paley conjecture that the coefficients of odd univalent functions are bounded by unity.
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