Geodesic
Means of the power $-2$ of derivatives in the class~$S$
Algebra i analiz, Tome 28 (2016) no. 6, pp. 189-207.

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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, $F$ does not maximize the expression $$\int _{|z|=r}\frac {1}{|F’(z)|^2} |dz| $$ in the class $S$.
Keywords: Brennan's conjecture, conformal mappings, means of the derivative of a conformal mapping, the class $S$. 
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N. A. Shirokov. Means of the power $-2$ of derivatives in the class~$S$. Algebra i analiz, Tome 28 (2016) no. 6, pp. 189-207. http://geodesic.mathdoc.fr/item/AA_2016_28_6_a7/

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