COEFFICIENT ESTIMATES FOR BI-UNIVALENT FUNCTIONS GENERATED BY SEMIGROUP OPERATORS

Abstract

The study of bi-univalent functions in the class Σ, where both the function and its inverse are univalent in the unit disk, continues to be a central problem in geometric function theory due to the intricate interdependence of their Taylor–Maclaurin coefficients. This paper investigates coefficient estimates for subclasses of bi-univalent functions generated by semigroup operators and subordination to various polynomial kernels, including generalized bivariate Fibonacci-like polynomials, (p,q)-Chebyshev polynomials of the second kind, Gegenbauer, and q-special functions. Using the Berkson–Porta representation of infinitesimal generators and techniques involving Schwarz functions, Carathéodory’s lemma, and algebraic systems derived from subordination conditions, sharp and non-sharp bounds are derived for the initial coefficients |a₂| (or |ε₂|) and |a₃|. The analysis incorporates q-calculus, fractional q-differential operators, and the Fekete–Szegő functional to explore how operator parameters, polynomial recursion, and quantum deformation influence coefficient growth. Historical bounds are reviewed, and modern generalizations demonstrate the unifying role of semigroup theory in bridging discrete and continuous transformations. The results reveal deeper connections between kernel complexity, operator order, and geometric constraints, offering new insights into deformation theory, k-fold symmetry, and fractal boundaries in analytic function classes.