Title: Univalent Biharmonic Mappings and Linearly Connected Domains
Author(s): Zayid Abdulhadi, L. El Hajj
Pages: 1-8
Cite as:
Zayid Abdulhadi, L. El Hajj, Univalent Biharmonic Mappings and Linearly Connected Domains, Int. J. Anal. Appl., 9 (1) (2015), 1-8.

Abstract


A four times continuously differentiable complex valued function F = u + iv in a simply connected domain Ω is biharmonic if the laplacian of F is harmonic. Every biharmonic mapping F in Ω has the representation F = |z|^2 G + K, where G and K are harmonic in Ω. This paper investigates the relationship between the univalence of F and of K using the concept of linearly connected domains.


Full Text: PDF

 

References


  1. Z. Abdulhadi and Y.Abumuhanna , “ Landau’s theorem for biharmonic mappings,” Journal of Mathematical Analysis and Applications, 338(2008), 705-709.

  2. Z. Abdulhadi, Y.Abumuhanna and S. Khoury, “ On Univalent Solutions of the Biharmonic Equations,” Journal of Inequalities and Applications, 2005(2005), 469-478.

  3. Z. Abdulhadi, Y.Abumuhanna and S. Khoury, “On the univalence of the log-biharmonic mappings” Journal of Mathematical Analysis and Applications, 289(2004), 629-638.

  4. Z. Abdulhadi, Y.Abumuhanna and S. Khoury, “ On some properties of solutions of the biharmonic Equations,” Appl. Math. Comput. 177(2006), 346-351.

  5. Y.Abu-Muhanna and R. M. Ali“ Biharmonic Maps and Laguerre Minimal Surfaces,” Journal of Abstract and Applied Analysis, 2013(2013), Article ID 843156, 9 pages.

  6. Y.Abu-Muhanna and G. Schober, Harmonic mappings onto convex mapping domains, Can. J. Math, 39(1987), 1489-1530.

  7. A. Bobenko and U. Pinkall, “Discrete isothermic surfaces,”Journal fur die Reine und Angewandte Mathematik, 475(1996), 187-208.

  8. W. Blaschke, “Uber die Geometrie von Laguerre III: Beitra ge ¨zur Fl achentheorie,” Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg ¨ , 4(1925), 1-12.

  9. W. Blaschke, “Uber die Geometrie von Laguerre II: Fla chen- ¨theorie in Ebenenkoordinaten,” Abhandlungen aus dem MathematischenSeminar der Universitat Hamburg ¨, 3(1924), 195- 212.

  10. S. Chen, S. Ponnusamy, and X. wang, “ Landau’s theorem for certain biharmonic mappings,” Applied Mathematics and Computation, 208(2009), 427-433.

  11. G. Choquet, Sur un type de transformation analytique g´en´eralisant la repr´esentation conforme et d´efinie au moyen de fonctions harmoniques, Bull. Sci. Math. 69(1945), 156–165.

  12. M.Chuaqui, R.Hermandez,Univalent Harmonic mappings and linearly connected domains, J. Math.Anal.Appl., 332(2007), 1189-1197.

  13. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Annales Acad. Sci. Fenn. Series A. Mathematica, 9(1984), 3-25.

  14. P. Duren, Harmonic mappings in the plane, Cambridge University Press, 2004.

  15. J. Happel and H. Brenner, Low reynolds Number Hydrodynamics, Pretice-Hall, 1965.

  16. W.E. Langlois, Slow Viscous Flow, Macmillan Company, 1964.

  17. M. Peternell and H. Pottmann, “A Laguerre geometric approachto rational offsets,” Computer Aided Geometric Design, 15(1998), 223-249.

  18. H. Pottmann, P. Grohs, and N. J. Mitra, “Laguerre minimal surfaces, isotropic geometry and linear elasticity,” Advances in Computational Mathematics, 31(2009), 391-419.