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Construction of planar harmonic functions. (English) Zbl 1139.31300
Summary: Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk can be written in the form $$f=h+\overline g$$, where $$h$$ and $$g$$ are analytic in the open unit disk. The functions $$h$$ and $$g$$ are called the analytic and coanalytic parts of $$f$$, respectively. In this paper, we construct certain planar harmonic maps either by varying the coanalytic parts of harmonic functions that are known to be harmonic starlike or by adjoining analytic univalent functions with coanalytic parts that are related or derived from the analytic parts.
##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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##### References:
 [1] J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, vol. 9, pp. 3-25, 1984. · Zbl 0506.30007 [2] P. Duren, Harmonic Mappings in the Plane, vol. 156 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, Mass, USA, 2004. · Zbl 1055.31001 [3] J. M. Jahangiri, C. Morgan, and T. J. Suffridge, “Construction of close-to-convex harmonic polynomials,” Complex Variables. Theory and Application, vol. 45, no. 4, pp. 319-326, 2001. · Zbl 1090.30501 [4] P. Greiner, “Geometric properties of harmonic shears,” Computational Methods and Function Theory, vol. 4, no. 1, pp. 77-96, 2004. · Zbl 1056.30023 · doi:10.1007/BF03321057 [5] J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470-477, 1999. · Zbl 0940.30003 · doi:10.1006/jmaa.1999.6377 [6] J. M. Jahangiri and H. Silverman, “Harmonic univalent functions with varying arguments,” International Journal of Applied Mathematics, vol. 8, no. 3, pp. 267-275, 2002. · Zbl 1026.30016 [7] H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 283-289, 1998. · Zbl 0908.30013 · doi:10.1006/jmaa.1997.5882 [8] H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of Mathematics, vol. 28, no. 2, pp. 275-284, 1999. · Zbl 0959.30003 [9] S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521-527, 1981. · Zbl 0458.30008 · doi:10.2307/2044151 [10] Y. Avci and E. Zlotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 44, pp. 1-7, 1990. · Zbl 0780.30013 [11] A. W. Goodman and E. B. Saff, “On univalent functions convex in one direction,” Proceedings of the American Mathematical Society, vol. 73, no. 2, pp. 183-187, 1979. · Zbl 0408.30010 · doi:10.2307/2042288
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