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In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes N Σ (γ , λ , δ , μ ; α) and N Σ * (γ , λ , δ , μ ; β) of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order μ using a convolution operator and the proofs of our results are based on the well-known Carathédory's inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions. [ABSTRACT FROM AUTHOR]