On the second Hankel determinant of concave functions

Abstract

In the present paper, we will discuss the Hankel determinants $H(f)=a_2{a}_4-a_3^2$ of order 2 for normalized concave functions $f(z)=z+a_2{z}^2+a_3{z}^3+\dots$ 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 $\phi (z)$ on $|z|<1$ with $|\phi |<1$ and $\phi (p)=p.$ 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 $p.$

10 pages