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.

Keywords

• analytic functions
• subordination
• Carathéodory functions
• sine function
• cosine function
• alpha-convex functions