The second Hankel determinant for starlike and convex functions of order alpha

Abstract

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions $f\in \mathcal{S}$ given by $f\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ for $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form $H_{2,2}\left(f\right)=a_2{a}_4-a_3^2$. A non-sharp bound for $H_{2,2}\left(f\right)$ when $f\in \mathcal{K}\left(\alpha \right)$, $\alpha \in [0,1)$ consisting of convex functions of order α was found by Krishna and Ramreddy (Hankel determinant for starlike and convex functions of order alpha. Tbil Math J. 2012;5:65–76), and later improved by Thomas et al. (Univalent functions: a primer. Berlin: De Gruyter; 2018). In this paper, we give the sharp result. Moreover, we obtain sharp results for $H_{2,2}\left(f^{-1}\right)$ for the inverse functions $f^{-1}$ when $f\in \mathcal{K}\left(\alpha \right)$, and when $f\in {\mathcal{S}}^{\ast }\left(\alpha \right)$, the class of starlike functions of order α. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and $f^{-1}$ for the classes ${\mathcal{S}}^{\ast }\left(\alpha \right)$, $\mathcal{K}\left(\alpha \right)$, ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$, where ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$ are, respectively, the classes of strongly starlike, and strongly convex functions of order β.

Keywords:

• Starlike functions
• convex functions
• coefficient problems
• Hankel determinant

AMS SUBJECT CLASSIFICATIONS:

• Primary: 30C45
• Secondary: 30C50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP; Ministry of Science, ICT and Future Planning) [grant number NRF-2017R1C1B5076778].