Geodesic
On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane
Dalʹnevostočnyj matematičeskij žurnal, Tome 6 (2005) no. 1, pp. 39-56.

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We study necessary and sufficient conditions for the preservation of the generalized reduced modulus under the extension of domain or a part of its boundary. Also, we give such conditions for polarization and dissymmetrization. As applications, the descriptions of extremal configurations in the known inequalities for the inner radii, the Green functions and the Robin functions are obtained. Some of our results supplement the known boundary distortion theorems for univalent regular functions. 
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V. N. Dubinin; E. G. Prilepkina. On the preservation of generalized reduced modulus under some geometric transformations of domains in the plane. Dalʹnevostočnyj matematičeskij žurnal, Tome 6 (2005) no. 1, pp. 39-56. http://geodesic.mathdoc.fr/item/DVMG_2005_6_1_a3/

[1] V. N. Dubinin, “Obobschennye kondensatory i asimptotika ikh emkostei pri vyrozhdenii nekotorykh plastin”, Zap. nauch. semin. POMI, 302, 2003, 38–51 | MR

[2] Dzh. Dzhenkins, Odnolistnye funktsii i konformnye otobrazheniya, Izd-vo inostr. lit., M., 1962

[3] K. Kheiman, Mnogolistnye funktsii, Izd-vo inostr. lit., M., 1960 | MR

[4] V. M. Miklyukov, “O nekotorykh granichnykh zadachakh teorii konformnykh otobrazhenii”, Sib. mat. zhurn., 18:5 (1977), 1111–1124 | MR | Zbl

[5] I. P. Mityuk, Simmetrizatsionnye metody i ikh primenenie v geometricheskoi teorii funktsii. Vvedenie v simmetrizatsionnye metody, Kubanskii gosuniversitet, Krasnodar, 1980

[6] P. Duren, M. M. Schiffer, “Robin functions and distortion of capacity under conformal mapping”, Complex Variables, 21 (1993), 189–196 | MR | Zbl

[7] V. N. Dubinin, “Simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo”, Uspekhi mat. nauk, 49:1 (1994), 3–76 | MR | Zbl

[8] G. V. Kuzmina, “Metody geometricheskoi teorii funktsii I, II”, Algebra i analiz, 9:3 (1997), 41–103 ; 5, 1–50 | MR | Zbl | MR | Zbl

[9] A. Yu. Solynin, “Moduli i ekstremalno-metricheskie problemy”, Algebra i analiz, 11:1 (1999), 3–86 | MR | Zbl

[10] P. Duren, “Robin capacity”, Computational methods and Function Theory, CMFT' 97, ed. N. Papamichael, St. Ruscheweyh, E. B. Saff, World scientific Publishing Co, 1999, 177–190 | MR | Zbl

[11] M. D. O'Neill, R. E. Thurman, “Extremal domains for Robin capacity”, Complex Variables, 41 (2000), 91–109 | MR

[12] R. W. Barnard, A. Yu. Solynin, “Local variations and minimal area problem for Caratheodory functions”, Indiana Univ. Math. J., 53:1 (2004), 135–167 | DOI | MR | Zbl

[13] S. Nasyrov, “Robin capacity and lift of infinitely thin airfoils”, Complex Variables, 47:2 (2002), 93–107 | MR | Zbl

[14] A. Vasil'ev, Moduli of families of curves for conformal and quasiconformal mappings, Lect. Notes in Math., 1788, 2002 | MR

[15] V. N. Dubinin, E. G. Prilepkina, “K teoremam edinstvennosti dlya preobrazovanii mnozhestv i kondensatorov na ploskosti”, Dalnevost. mat. zhurn., 3:2 (2002), 137–149 | MR

[16] A. Yu. Solynin, “Polyarizatsiya i funktsionalnye neravenstva”, Algebra i analiz, 8:6 (1996), 148–185 | MR | Zbl

[17] V. A. Shlyk, “O teoreme edinstvennosti dlya simmetrizatsii proizvolnykh kondensatorov”, Sib. mat. zhurn., 1982, no. 2, 165–175 | MR | Zbl

[18] V. N. Dubinin, Emkosti kondensatorov v geometricheskoi teorii funktsii, Izd-vo Dalnevost. un-ta, Vladivostok, 2003

[19] V. N. Dubinin, L. V. Kovalev, “Privedennyi modul kompleksnoi sfery”, Zap. nauch. semin. POMI, 254, 1998, 76–94 | MR

[20] L. V. Kovalev, “Monotonnost obobschennogo privedennogo modulya”, Zap. nauch. semin. POMI, 276, 2001, 219–236 | MR | Zbl

[21] D. Betsakos, “Polarization, conformal invariants, and Brownian motion”, Ann. Acad. Sci. Fen. Math., 23 (1998), 59–82 | MR | Zbl

[22] V. N. Dubinin, “Asimptotika modulya vyrozhdayuschegosya kondensatora i nekotorye ee primeneniya”, Zap. nauch. semin. POMI, 237, 1997, 56–73 | MR | Zbl

[23] V. N. Dubinin, N. V. Eirikh, “Nekotorye primeneniya obobschennykh kondensatorov v teorii analiticheskikh funktsii”, Zap. nauch. cemin. POMI, 314, 2004, 52–75 | MR | Zbl

[24] A. Yu. Solynin, “Granichnoe iskazhenie i ekstremalnye zadachi v nekotorykh klassakh odnolistnykh funktsii”, Zap. nauch. semin. POMI, 204, 1993, 115–142

[25] Ch. Pommerenke, A. Vasil'ev, “Angular derivatives of bounded univalent functions and extremal partions of the unit disk”, Pacific J. Math., 206:2 (2002), 425–450 | DOI | MR | Zbl

[26] Pommerenke Ch., Boundary behaviour of conformal maps, Springer, Berlin, 1992 | MR | Zbl

[27] R. W. Barnard, K. Pearce, A. Yu. Solynin, “Area, width, and logarithmic capacity of convex sets”, Pacific J. Math., 212:1 (2003), 13–23 | DOI | MR | Zbl

[28] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1973 | MR