Skip to main content
Log in

Properties of Harmonic Functions Defined by Shear Construction

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3–25 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Droff, M.: Convolutions of plannar harmonic convex mappings. Complex Var. Theory Appl. 45, 263–271 (2001)

    Google Scholar 

  3. Goodman, A.W.: Univalent functions and nonanalytic curves. Proc. Am. Math. Soc. 8, 598–601 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  5. Jahangiri, J.: Harmonic functions starlike in the unit disk. J. Math. Anal. Appl. 235, 470–477 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  7. Libera, R.J.: Some classes of regular univalent functions. Proc. Am. Math. Soc. 16, 755–758 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Muir, S.: Weak subordination for convex univalent harmonic function. J. Math. Anal. Appl. 348, 862–871 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Muir, S.: Harmonic mappings convex in one or every direction. Comput. Methods Funct. Theory. 12(1), 221–239 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pokhrel, C.M.: Convexity preservation for analytic, harmonic and plane curves. Ph.D. thesis. Tribhuvan University, Kathmandu (2004)

    Google Scholar 

  11. Pommenrenke, C.: On starlike and close-to-convex functions. Proc. London Math. Soc. 13, 290–304 (1963)

    Article  MathSciNet  Google Scholar 

  12. Ruscheweyh, S., Sheil-Small, T.: Hadamard products of Schlicht functions and the Polya Schoenberg conjecture. Comment. Math. Helv. 48, 119–135 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ruscheweyh, S., Salinas, L.: On the preservation of direction-convexity and the Goodman-Saff conjecture. Ann. Acad. Sci. Fenn. 14, 63–73 (1989)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rasoul Aghalary.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghalary, R., Rad, M.J. Properties of Harmonic Functions Defined by Shear Construction. Acta Math Vietnam 43, 471–483 (2018). https://doi.org/10.1007/s40306-017-0235-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-017-0235-y

Keywords

Mathematics Subject Classification (2010)

Navigation