Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective

Abstract

This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results.

New Uses of q-Generalized Janowski Function in q-Bounded Turning Functions

“…Researchers in the field have the opportunity to investigate this problem and utilize recent established operators like [40,41] in geometric function theory on bounded turning function or other subclasses like starlike function, convex functions, and others, too, to uncover additional properties like the third Hankel determinant [39].…”