Applications of Zalcman Conjecture for Analytic Functions Linked to Cosine Function

In this paper, we investigate applications of the Zalcman conjecture for analytic functions associated with the cosine function. We introduce and analyze a new class of normalized analytic functions within the open unit disk, leveraging subordination principles. The coefficient estimates for this class are derived, and explicit upper bounds for the first five coefficients are established. Furthermore, we prove the validity of the Zalcman conjecture for this function class by obtaining an upper bound for the Zalcman functional. Our results contribute to the broader study of analytic function theory and extend existing findings on function classes linked to trigonometric functions.