Geodesic
On Mappings Related to the Gradient of the Conformal Radius
Matematičeskie zametki, Tome 87 (2010) no. 1, pp. 3-12.

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We establish a criterion for the gradient $\nabla R(D,z)$ of the conformal radius of a convex domain $D$ to be conformal: the boundary $\partial D$ must be a circle. We obtain estimates for the coefficients $K(r)$ for the $K(r)$-quasiconformal mappings $\nabla R(D,z)$, $D(r)\subset D$, $0$, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain $D$.
Keywords: conformal radius, gradient of the conformal radius, convex mapping
Mots-clés : coefficient of quasiconformality, astroid, cycloid, hypocycloid. 
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L. A. Aksent'ev; A. N. Akhmetova. On Mappings Related to the Gradient of the Conformal Radius. Matematičeskie zametki, Tome 87 (2010) no. 1, pp. 3-12. http://geodesic.mathdoc.fr/item/MZM_2010_87_1_a0/

[1] G. Polia, G. Sege, Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatlit, M., 1962 | MR | Zbl

[2] F. G. Avkhadiev, K.-J. Wirths, Schwarz–Pick Type Inequalities, Front. Math., Birkhäuser Verlag, Basel, 2009 | MR | Zbl

[3] H. R. Haegi, “Extremalprobleme und Ungleichungen konformer Gebietsgrössen”, Compositio Math., 8 (1950), 81–111 | MR | Zbl

[4] F. G. Avkhadiev, Konformno invariantnye neravenstva i ikh prilozheniya, Preprint No 95-1, Kazanskii fond “Matematika”, Kazan, 1995

[5] F. G. Avkhadiev, K.-J. Wirths, “The conformal radius as a function and its gradient image”, Israel J. Math., 145 (2005), 349–374 | DOI | MR | Zbl

[6] A. A. Savelov, Ploskie krivye. Sistematika, svoistva, primeneniya, Fizmatgiz, M., 1960 | MR

[7] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987 | MR | Zbl

[8] L. Alfors, Lektsii po kvazikonformnym otobrazheniyam, Mir, M., 1969 | MR | Zbl

[9] L. A. Aksentev, “Lokalnoe stroenie poverkhnosti vnutrennego konformnogo radiusa dlya ploskoi oblasti”, Izv. vuzov. Matem., 2002, no. 4, 3–12 | MR | Zbl