On the second Hankel determinant of concave functions

TL;DR

This paper investigates the second Hankel determinant for normalized concave functions with a pole, characterizing coefficient bounds and contributing to extremal problems in geometric function theory.

Contribution

It characterizes the coefficient body of order 2 for a class of meromorphic functions with a pole, aiding in extremal problems for concave functions.

Findings

Characterization of the coefficient body for functions with a pole at p

Analysis of the Hankel determinant H(f) for concave functions

Potential applications to extremal coefficient problems

Abstract

In the present paper, we will discuss the Hankel determinants $H(f) =a_2a_4-a_3^2$ of order 2 for normalized concave functions $f(z)=z+a_2z^2+a_3z^3+\dots$ with a pole at $p\in(0,1).$ Here, a meromorphic function is called concave if it maps the unit disk conformally onto a domain whose complement is convex. To this end, we will characterize the coefficient body of order 2 for the class of analytic functions $\varphi(z)$ on $|z|<1$ with $|\varphi|<1$ and $\varphi(p)=p.$ We believe that this is helpful for other extremal problems concerning $a_2, a_3, a_4$ for normalized concave functions with a pole at $p.$