On means with power $-2$ for the dervatives of functions of class $S$

Abstract:

Let $S$ be the standard class of conformal mapping of the unit disk $\mathbb D$, and let $F\in \mathbb D$. Suppose that there exist Jordan domains $G_1$ and $G$, $G_1\supset G$, such that $G\subset \mathbb C\setminus f(\mathbb D)$, $\partial f(\mathbb D)\cap \partial G$ contains a Dini-smoth arc $\gamma$, and $G_1 \cap \partial f(\mathbb D) \cap \partial G=\gamma$. It is established that, in this case, for any $r$ with $0<r<1$, $F$ does not maximize the expression \begin{equation*} \int _{|z|=r}\frac {1}{|F’(z)|^2} |dz| \end{equation*} in the class $S$.