Fekete–Szegö inequalities for bi-univalent functions involving cauchy and Bernoulli polynomials

We present a unified complex-variable extension of both Cauchy and Bernoulli polynomials on the open unit disc. This generalized framework deepens their analytic properties and opens new directions for application in geometric function theory. Within this setting, we introduce and analyze two bi-univalent function classes, for which we establish upper bounds for the initial Taylor coefficients |a2| and |a3|, along with refined Fekete-Szegoe-type inequalities. Starlike and convex subclasses naturally emerge as special cases of the general theory. To the best of our knowledge, this is the first work to jointly apply Cauchy and Bernoulli polynomials in the study of bi-univalent functions, laying the groundwork for future