Abstract
Given a convex complex-valued analytic mapping on the open unit disk in ℂ, we construct a family of complex-valued harmonic mappings convex in the direction of the imaginary axis. We also show that adding the condition of direction convexity preserving to the analytic mapping is a necessary and sufficient condition for the harmonic mapping to be convex. Using analytic radial slit mappings, we provide a three parameter family of harmonic mappings convex in the direction of the imaginary axis and show in some cases that as one parameter varies continuously, the mappings vary from being convex in the direction of the imaginary axis to being convex. In so doing, we also provide information on whether or not some analytic mappings are direction convexity preserving. Lastly, we will provide coefficient conditions leading to harmonic mappings which are starlike or convex of order α.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. 17 (1915–1916), 12–22.
J. Clunie and T. Sheil-Small, Harmonic univalent functions, Annales Acad. Sci. Fenn. 9 (1984), 3–25.
A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601.
J. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect. A 2(1998), 57–66.
J. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235 (1999), 470–477.
H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692.
S. Muir, Weak subordination for convex univalent harmonic functions, J. Math. Anal. Appl. 348 (2008), 862–871.
C. Pokhrel, On a subclass of DCP, Nepali Math Sci. Rep. 22 (2004), 67–85.
S. Ruscheweyh and L. Salinas, On the DCP radius of exponential partial sums, J. Comput. Anal. Appl. 7 (2005), 271–288.
S. Ruscheweyh and L. Salinas, On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. 14 (1989), 63–73.
S. Ruscheweyh and T. Suffridge, A continuous extension of the de la Vallée Poussin means, J. Anal. Math. 89 (2003), 155–167.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Muir, S. Harmonic Mappings Convex in One or Every Direction. Comput. Methods Funct. Theory 12, 221–239 (2012). https://doi.org/10.1007/BF03321824
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321824
Keywords
- Harmonic mappings
- convex
- convex in one direction
- direction convexity preserving
- Hadamard product
- order of convexity
- order of starlikeness