Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON CONVEXITY OF CONVOLUTIONS OF HARMONIC MAPPINGS

  • JIANG, YUE-PING (School of Mathematics and Econometrics Hunan University) ;
  • RASILA, ANTTI (Department of Mathematics and Systems Analysis Aalto University) ;
  • SUN, YONG (School of Mathematics and Econometrics Hunan University)
  • Received : 2014.08.19
  • Published : 2015.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study right half-plane harmonic mappings $f_0$ and f, where $f_0$ is fIxed and f is such that its dilatation of a conformal automorphism of the unit disk. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy [11], which itself originates from a problem posed by Dorff et al. in [7].

Keywords

References

  1. O. P. Ahuja and J. M. Jahangiri, Convolutions for special classes of harmonic univalent functions, Appl. Math. Lett. 16 (2003), no. 6, 905-909. https://doi.org/10.1016/S0893-9659(03)90015-2
  2. R. M. Ali, B. A. Stephen, and K. G. Subramanian, Subclass of harmonic mappings de ned by convolution, Appl. Math. Lett. 23 (2010), no. 10, 1243-1247. https://doi.org/10.1016/j.aml.2010.06.006
  3. D. Bshouty and A. Lyzzaik, Problems and conjectures in planar harmonic mappings, J. Anal. 18 (2010), 69-81.
  4. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9 (1984), 3-25. https://doi.org/10.5186/aasfm.1984.0905
  5. A. Cohn, Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), no. 1, 110-148. https://doi.org/10.1007/BF01215894
  6. M. Dorff, Convolutions of planar harmonic convex mappings, Complex Var. Theory Appl. 45 (2001), no. 3, 263-271. https://doi.org/10.1080/17476930108815381
  7. M. Dorff, M. Nowak, and M.Wo loszkiewicz, Convolutions of harmonic convex mappings, Complex Var. Elliptic Equ. 57 (2012), no. 5, 489-503. https://doi.org/10.1080/17476933.2010.487211
  8. P. Duren, Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004.
  9. M. Goodloe, Hadamard products of convex harmonic mappings, Complex Var. Theory Appl. 47 (2002), no. 2, 81-92. https://doi.org/10.1080/02781070290010841
  10. R. Kumar, S. Gupta, S. Singh, and M. Dor , On harmonic convolutions involving a vertical strip mapping, Bull. Korean Math. Soc. 52 (2015), no. 1, 105-123. https://doi.org/10.4134/BKMS.2015.52.1.105
  11. L.-L. Li and S. Ponnusamy, Solution to an open problem on convolutions of harmonic mappings, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1647-1653. https://doi.org/10.1080/17476933.2012.702111
  12. L.-L. Li and S. Ponnusamy, Convolutions of slanted half-plane harmonic mappings, Analysis (Munich) 33 (2013), no. 2, 159-176. https://doi.org/10.1524/anly.2013.1170

Cited by

  1. Univalency of Convolutions of Univalent Harmonic Right Half-Plane Mappings vol.17, pp.2, 2017, https://doi.org/10.1007/s40315-016-0180-0