The sharp bounds of the second and third Hankel determinants for the class SL^*

TL;DR

This paper determines the exact bounds of specific Hankel determinants and the Zalcman functional for the class SL^* of starlike functions related to the right lemniscate, advancing geometric function theory.

Contribution

It provides the sharp bounds for H_2(3), H_3(1), and the Zalcman functional for the class SL^*, with additional new results on this class.

Findings

Sharp bounds for H_2(3) and H_3(1) determinants.

Exact bound for the Zalcman functional when n=3.

Additional interesting properties of the class SL^*.

Abstract

The aim of the present paper is to obtain the sharp bounds of the Hankel determinants H_2(3) and H_3(1) for the well known class SL^* of starlike functions associated with the right lemniscate of Bernoulli. Further for n=3, we find the sharp bound of the Zalcman functional for the class SL^*. In addition, a couple of interesting results of SL^* is appended at the end.