In this work, by using the Al-Oboudi differential operator and the rule of subordination, we introduced the new subclasses D
n,ρ
Σ,
δ
(Φ) and F
n,α
Σ,
δ
(Φ) of the bi-univalent functions. Likewise, we use the Fibonacci numbers to derive the initial coefficients bounds for |a
2| and |a
3| of the bi-univalent function subclasses.
Let S denote the class of analytic and univalent functions in D, where D is defined as unit disk and having the Taylor representation form of S. We will determine the estimation for the Toeplitz determinants where the elements are the Taylor coefficients of the class close-to-convex functions in S.
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