Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:03:07.032Z Has data issue: false hasContentIssue false
  • English
  • Français

RADII OF HARMONIC MAPPINGS IN THE PLANE

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  08 July 2016

BO-YONG LONG  and
HUA-YING HUANG
BO-YONG LONG*
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, China email longboyong@ahu.edu.cn
HUA-YING HUANG
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, China email huanghuaying50@163.com
*
✉longboyong@ahu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

Keywords

harmonic mappingsradiusfully convex of order 𝛼fully starlike of order 𝛼convex combinationconvolutionBloch constant

MSC classification

Primary: 31A05: Harmonic, subharmonic, superharmonic functions
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 331 - 347
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The research was supported by the NSFC (no. 11501002), Doctoral Research Start-up Funds Projects of Anhui University (numbers J10113190002 and 01001901) and partially by China Scholarship Council.

References

Ahuja, O. P., Nagpal, S. and Ravichandran, V., ‘Radius constants for functions with the prescribed coefficient bounds’, Abstr. Appl. Anal. 2014 (2014), article ID 454152, 12 pages.CrossRefGoogle Scholar
Chen, H., Gauthier, P. M. and Hengartner, W., ‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc. 128(11) (2000), 3231–3240.CrossRefGoogle Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Bloch constant and Landau’s theorems for planar p-harmonic mappings’, J. Math. Anal. Appl. 373(2011) 102–110.CrossRefGoogle Scholar
Chuaqui, M., Duren, P. and Osgood, B., ‘Curvature properties of planar harmonic mappings’, Comput. Methods Funct. Theory 4(1) (2004), 127–142.CrossRefGoogle Scholar
Clunie, J. and Sheil-Small, T., ‘Harmonic univalent functions’, Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25.CrossRefGoogle Scholar
Dorff, M., ‘Convolutions of planar harmonic convex mappins’, Complex Var. Theory Appl. 45(3) (2001), 263–271.Google Scholar
Dorff, M. and Nowak, M., ‘Landau’s theorem for planar harmonic mappings’, Comput. Methods Funct. Theory 4(1) (2004), 151–158.CrossRefGoogle Scholar
Goodloe, M. R., ‘Hadamard products of convex harmonic mappings’, Complex Var. Theory Appl. 47(2) (2002), 81–92.Google Scholar
Jahangiri, J. M., ‘Coefficient bounds and univalence criteria for harmonic functions with negative coefficients’, Ann. Univ. Mariae Curie-Skłodowska Sect. A 52(2) (1998), 57–66.Google Scholar
Jahangiri, J. M., ‘Harmonic functions starlike in the unit disk’, J. Math. Anal. Appl. 235(2) (1999), 470–477.CrossRefGoogle Scholar
Kalaj, D., Ponnusamy, S. and Vuorinen, M., ‘Radius of close-to-convexity of harmonic functions’, Complex Var. Elliptic Equ. 59(4) (2014), 539–552.CrossRefGoogle Scholar
Lewy, H., ‘On the non-vanishing of the Jacobian in certain one-to-one mappings’, Bull. Amer. Math. Soc. 42(10) (1936), 689–692.CrossRefGoogle Scholar
Liu, M.-Sh., ‘Estimates on Bloch constants for planar harmonic mappings’, Sci. China Ser. A 52(1) (2009), 87–93.CrossRefGoogle Scholar
MacGregor, T. H., ‘The univalence of a linear combination of convex mappings’, J. Lond. Math. Soc. 44 (1969), 210–212.CrossRefGoogle Scholar
Nagpal, S. and Ravichandran, V., ‘Fully starlike and convex harmonic mappings of order 𝛼’, Ann. Polon. Math. 108(1) (2013), 85–107.CrossRefGoogle Scholar
Ruscheweyh, S. and Salinas, L., ‘On the preservation of direction-convexity and the Goodman–Saff conjecture’, Ann. Acad. Sci. Fenn. Ser. A. I Math. 14(1989) 63–73.CrossRefGoogle Scholar
Sheil-Small, T., ‘Constants for planar harmonic mappings’, J. Lond. Math. Soc. 42(2) (1990), 237–248.CrossRefGoogle Scholar
Wang, X.-T., Liang, X.-Q. and Zhang, Y.-L., ‘Precise coefficient estimates for close-to-convex harmonic univalent mappings’, J. Math. Anal. Appl. 263(2) (2001), 501–509.CrossRefGoogle Scholar