Acta Mathematica Vietnamica

, Volume 43, Issue 3, pp 471–483 | Cite as

Properties of Harmonic Functions Defined by Shear Construction

  • Rasoul Aghalary
  • Mahdi Jahani Rad


In this paper, we introduce a new subclass of harmonic functions defined by shear construction. Among other properties of this subclass, we study the convolution of its elements of it with some special subclasses of harmonic functions. Also, we provide coefficient conditions leading to harmonic mappings which are starlike of order \(\alpha \). Some interesting applications of the general results are also presented.


Harmonic starlike and convex functions Shear construction Univalent harmonic function Radii problem 

Mathematics Subject Classification (2010)

Primary 30C45 Secondary 30C80 



The authors would like to thank the referees for their careful reading of the paper and for their helpful comments to improve it.


  1. 1.
    Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3–25 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Droff, M.: Convolutions of plannar harmonic convex mappings. Complex Var. Theory Appl. 45, 263–271 (2001)Google Scholar
  3. 3.
    Goodman, A.W.: Univalent functions and nonanalytic curves. Proc. Am. Math. Soc. 8, 598–601 (1957)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Jahangiri, J.: Coefficient bounds and univalence criteria for harmonic functions with negative coefficient. Ann. Univ. Mariae Curie-Sklodowska Sect. A. 2, 57–66 (1998)MathSciNetMATHGoogle Scholar
  5. 5.
    Jahangiri, J.: Harmonic functions starlike in the unit disk. J. Math. Anal. Appl. 235, 470–477 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Libera, R.J.: Some classes of regular univalent functions. Proc. Am. Math. Soc. 16, 755–758 (1965)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Muir, S.: Weak subordination for convex univalent harmonic function. J. Math. Anal. Appl. 348, 862–871 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Muir, S.: Harmonic mappings convex in one or every direction. Comput. Methods Funct. Theory. 12(1), 221–239 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pokhrel, C.M.: Convexity preservation for analytic, harmonic and plane curves. Ph.D. thesis. Tribhuvan University, Kathmandu (2004)Google Scholar
  11. 11.
    Pommenrenke, C.: On starlike and close-to-convex functions. Proc. London Math. Soc. 13, 290–304 (1963)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ruscheweyh, S., Sheil-Small, T.: Hadamard products of Schlicht functions and the Polya Schoenberg conjecture. Comment. Math. Helv. 48, 119–135 (1973)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ruscheweyh, S., Salinas, L.: On the preservation of direction-convexity and the Goodman-Saff conjecture. Ann. Acad. Sci. Fenn. 14, 63–73 (1989)MathSciNetMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUrmia UniversityUrmiaIran

Personalised recommendations