Multivalent Harmonic Uniformly Starlike Functions

  • Ahuja, Om (Department of Mathematics Kent State University) ;
  • Joshi, Santosh (Department of Mathematics Walchand College of Engineering) ;
  • Sangle, Naveneet (Department of Mathematics Annasaheb Dange College of Engineering)
  • Received : 2008.07.17
  • Accepted : 2008.09.12
  • Published : 2009.09.30


In this paper, we investigate a generalized family of complex-valued harmonic functions that are multivalent, sense-preserving, and are associated with k-uniformly harmonic functions in the unit disk. The results obtained here include a number of known and new results as their special cases.


harmonic functions;multivalent harmonic functions;convex combinations;distortion bounds;uniformly harmonic starlike


  1. O. P. Ahuja, R. Aghalary and S. B. Joshi, Harmonic univalent functions associated with k-uniformly starlike functions, Math. Sci. Res. J., 9(1)(2005), 9-17.
  2. O. P. Ahuja and J. M. Jahangiri, Noshiro-type harmonic univalent functions, Scientiae Math. Japonicae, 56(2)(2002), 293-299.
  3. O. P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions with missing coecients, Math. Sci. Res. J., 7(9)(2003), 347-352.
  4. O. P. Ahuja and J. M. Jahangiri, On a linear combination of classes of multivalently harmonic functions, Kyungpook Math. J., 42(1)(2002), 61-70.
  5. O. P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions, Ann. Univ. Mariae Curie-Sklodowska, Vol. LV, 1 Sectio A, 55(1)(2001), 1-13.
  6. O. P. Ahuja and J. M. Jahangiri, Errata to "Multivalent harmonic starlike functions" [Univ. Mariae Curie-Sklodowska, Vol LV, sectio A, 55(1)(2001), 1-13]
  7. O. P. Ahuja and J. M. Jahangiri, Errata to "Multivalent harmonic starlike functions" Univ. Mariae Curie-Sklodowska, Sectio. A, 56(1)(2002), 105.
  8. M. K. Aouf, A generalization of multivalent functions with negative coefficient II, Bull Korean Math. Soc., 25(2)(1988), 221-232.
  9. R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math., 25(1997), 17-32.
  10. D. Bshouty, W. Hengartner and M. Naghibi-Beidokhti, p-valent harmonic mapping with finite Blaschke dilatations, Ann. Univ. Mariae Curie. Sklodowska Sectio. A, 53(2)(1999), 9-26.
  11. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acd. Aci. Fenn. Ser. A I Math., 9(1984), 3-25.
  12. P. L. Duren, Harmonic mappings in the plane, Cambridge Tract in Mathematics (Cambridge: 156, Cambridge University Press), 2004, ISBN: 0-521-64121-73.
  13. P. L. Duren, W. Hengartner and R. S. Laugesen, The argument principles harmonic functions, Amer. Math. Monthly, 103(5)(1996), 411-415.
  14. A. W. Goodman, On uniformly convex functions, Ann. Polon, Math., 56(1)(1991), 87-92.
  15. A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl., 155(1991), 364-370.
  16. H. O. Guney and O. P. Ahuja, Inequalities involving multipliers for multivalent harmonic functions, J. Ineq. Pure Appl. Math., 7(5), Article 190, (2006), 1-9.
  17. J. M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math. Anal. Appl., 235(1999), 470-477.
  18. J. M. Jahangiri, G. Murugusundaramoorthy and K. Vijaya, On starlikeness of certain multivalent Harmonic functions, Journal of Natural Geometry, 24(2003), 1-10.
  19. S. Kanas and H. M. Srivastava, Linear operators associated with k-uniformly convex functions, Integral Transform. Spect. Funct., 9(2)(2000), 121-132.
  20. W. Ma and D. Minda, Uniformly convex functions, Ann Polon Math., 57(1992), 165-175.
  21. J. Patel, On a class of p-valent functions with negative and missing coefficients, Kyungpook Math. J., 36(1996), 29-40.
  22. F. Ronning, On uniform starlikeness and related properties of univalent functions, Complex variables Theory Appl., 24(1994), 233-239.
  23. F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118(1993), 189-196.
  24. T. Rosy, B. A, Stephen, K. G. Subramanian and J. M. Jahangiri, Goodman-Ronning-Type harmonic univalent functions, Kyungpook Math. J., 41(2001), 45-54.
  25. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220(1998), 283-289.
  26. H. Silverman and E. M. Silvia, Subclasses of harmonic univalent functions, New Zealand J. Math., 28(1999), 275-284.