On the second Hankel determinant of concave functions

Abstract: In the present paper, we will discuss the Hankel determinants $H(f) =a_2a_4-a_3<sup>2$ of order 2 for normalized concave functions $f(z)=z+a_2z<sup>2+a_3z<sup>3+\dots$ 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 $|z|&lt;1$ with $|\varphi|&lt;1$ 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 $p.$