A study of sharp coefficient bounds for a new subfamily of starlike functions

Abstract

Abstract In this article, by employing the hyperbolic tangent function tanhz, a subfamily S tanh ∗ $\mathcal{S}_{\tanh }^{\ast }$ of starlike functions in the open unit disk D ⊂ C $\mathbb{D}\subset \mathbb{C}$ : D = { z : z ∈ C and | z | < 1 } $$\begin{aligned} \mathbb{D}= \bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1 \bigr\} \end{aligned}$$ is introduced and investigated. The main contribution of this article includes derivations of sharp inequalities involving the Taylor–Maclaurin coefficients for functions belonging to the class S tanh ∗ $\mathcal{S}_{\tanh }^{\ast } $ of starlike functions in D $\mathbb{D}$ . In particular, the bounds of the first three Taylor–Maclaurin coefficients, the estimates of the Fekete–Szegö type functionals, and the estimates of the second- and third-order Hankel determinants are the main problems that are proposed to be studied here.

Keywords

• Analytic (or regular or holomorphic) functions
• Univalent functions
• Starlike functions
• Principle of subordination
• Schwarz function
• Hyperbolic and trigonometric functions