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In this work, we consider certain class of bi-univalent functions related with shell-like curves related to κ−Fibonacci numbers. Further, we obtain the estimates of initial Taylor-Maclaurin coefficients (second and third coefficients) and Fekete -Szegö inequalities. Also we discuss the special cases of the obtained results.

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In this paper, by making use of Borel distribution we introduce a new family GΣ(δ, γ, λ, τ, r) of normalized analytic and bi-univalent functions in the open unit disk U, which are associated with Horadam polynomials. We establish upper bounds for the initial Taylor-Maclaurin coefficients |a2| and |a3| of functions belonging to the analytic and bi-univalent function family which we have introduced here. Furthermore, we establish the Fekete-Szego problem of functions in this new family.

The object of the present paper is to derive some properties of holomorphic functions in the open unit disc which are defined by using a new generalized integral operator by applying a lemma due to Miller and Mocanu. Also we mention some interesting consequences of our main results.