Fekete–Szegő and Zalcman Functional Estimates for Subclasses of Alpha-Convex Functions Related to Trigonometric Functions

Abstract

In this study, we introduce the new subclasses, Mα(sin) and Mα(cos), of α-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class Mα(cos) for the case n=3, proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied.

Subclasses of Analytic Functions Subordinated to the Four-Leaf Function

“…Many results regarding some subclasses defined by subordinations with different functions with significant geometrical properties (e.g., the limaçon function, convex func-tions in one direction, the cosine function, the nephroid function, etc.) were studied by the fourth author in many papers (see, for example, [29][30][31][32]). The novelty of these subclasses and of this paper consists in the fact that such subordinations with similar expressions to the left-hand side of the subordination (4) were not studied in some other previous articles.…”