Properties of Harmonic Functions Defined by Shear Construction


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.

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  1. 1.

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

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

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Correspondence to Rasoul Aghalary.

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Aghalary, R., Rad, M.J. Properties of Harmonic Functions Defined by Shear Construction. Acta Math Vietnam 43, 471–483 (2018).

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

Mathematics Subject Classification (2010)

  • Primary 30C45
  • Secondary 30C80