Condenser Capacity Under Multivalent Holomorphic Functions

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• 作者：Michael Papadimitrakis (1)
  Stamatis Pouliasis (2) (3)
  
• 关键词：Condenser capacity ; Green energy ; Valency ; Quasiregular mappings ; Primary 30C85 ; Secondary 30C80 ; 31A15
• 刊名：Computational Methods and Function Theory
• 出版年：2013
• 出版时间：May 2013
• 年：2013
• 卷：13
• 期：1
• 页码：11-20
• 参考文献：1. Dubinin, V.N.: A majorization principle for p-valent functions. Mat. Zametki 65(4), 533鈥?41 (1999) (Russian), translation in Math. Notes 65(3鈥?), 447鈥?53 (1999)
  2. Dubinin, V.N.: On the preservation of conformal capacity under meromorphic functions. Analytical theory of numbers and theory of functions. Part 26, Zap. Nauchn. Sem. POMI, 392, POMI, St. Petersburg, pp 67鈥?3 (2011) (Russian), translation in J. Math. Sci. (NY) 184(6), 699鈥?02 (2012)
  3. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
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  7. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972) CrossRef
  8. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, Second Edition, Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer, Berlin (1973)
  9. Martio, O.: A capacity inequality for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I(474), 1鈥?8 (1970)
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  11. Pouliasis, S.: Condenser capacity and meromorphic functions. Comput. Methods Funct. Theory 11(1), 237鈥?45 (2011) CrossRef
  12. Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995) CrossRef
  13. Rickman, S.: Quasiregular Mappings, Results in Mathematics and Related Areas (3), vol. 26. Springer, Berlin (1993) CrossRef
  14. Sevost鈥檡anov, E.: The V盲is盲l盲 inequality for mappings with finite length distortion. Complex Var. Elliptic Equ. 55(1鈥?), 91鈥?01 (2010)
  15. Srivastava, S.M.: A Course on Borel Sets, Graduate Texts in Mathematics, vol 180. Springer, Berlin (1998) CrossRef
  16. V盲is盲l盲, J.: Modulus and capacity inequalities for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I(509), 1鈥?4 (1972)
  
• 作者单位：Michael Papadimitrakis (1)
  Stamatis Pouliasis (2) (3)
  
  1. Department of Mathematics, University of Crete, Knossos Ave, Iraklio, 71409, Greece
  2. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, 541242, Greece
  3. D茅partement de Math茅matiqes et de Statistique, Universit茅 Laval, Quebec, G1V 0A6, QC, Canada
  
• ISSN：2195-3724

文摘

We prove an inequality for the capacity of a condenser via a holomorphic function f, under a valency assumption on f, and we show that equality occurs if and only if f has finite constant valency. Also, we generalize a well known inequality for quasiregular mappings and we give a necessary condition for the case of equality.