Coefficient Bounds for New Subclasses of Bi-Univalent Functions Involving the S\(\breve{a}\)l\(\breve{a}\)gean Deferential Operator and Fekete–Szeg\(\ddot{o}\) Inequalities
A. Naik *
P. G. Department of Mathematics, Fakir Mohan University, Balasore-756019, Odisha, India.
S.C. Sahoo
P. G. Department of Mathematics, Fakir Mohan University, Balasore-756019, Odisha, India.
*Author to whom correspondence should be addressed.
Abstract
The study of bi-univalent functions has attracted considerable attention in geometric function theory due to its importance in coefficient problems and related applications. Motivated by recent developments in differential operators, this paper introduces two new subclasses of bi-univalent functions belonging to the family Σ in the open unit disk using the S˘al˘agean differential operator. For these subclasses, upper bounds for the initial coefficients |b2| and |b3| are obtained. The results extend and unify several earlier findings in the literature. In addition, a Fekete-Szeg¨o type inequality is established for the newly defined subclasses. The proposed framework provides a useful basis for further investigations, including higher-order coefficient estimates, Hankel determinants, and related problems.
Keywords: Analytic and univalent mappings, Bi-univalent functions, Starlike and convex subclasses, coefficient inequalities, S˘al˘agean operator