On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

Agnieszka Sibelska

Annales UMCS, Mathematica (2010)

  • Volume: 64, Issue: 1, page 81-91
  • ISSN: 2083-7402

Abstract

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The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998-2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6], among others. This work continues the investigations described in [7]. Our results relate primarily to the order of starlikeness and convexity of functions of the aforementioned classes.

How to cite

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Agnieszka Sibelska. "On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition." Annales UMCS, Mathematica 64.1 (2010): 81-91. <http://eudml.org/doc/268102>.

@article{AgnieszkaSibelska2010,
abstract = {The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998-2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6], among others. This work continues the investigations described in [7]. Our results relate primarily to the order of starlikeness and convexity of functions of the aforementioned classes.},
author = {Agnieszka Sibelska},
journal = {Annales UMCS, Mathematica},
keywords = {Complex harmonic functions; analytic conditions; convexity of order β; starlikeness of order β; complex harmonic functions; convexity of order ; starlikeness of order },
language = {eng},
number = {1},
pages = {81-91},
title = {On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition},
url = {http://eudml.org/doc/268102},
volume = {64},
year = {2010},
}

TY - JOUR
AU - Agnieszka Sibelska
TI - On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition
JO - Annales UMCS, Mathematica
PY - 2010
VL - 64
IS - 1
SP - 81
EP - 91
AB - The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998-2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6], among others. This work continues the investigations described in [7]. Our results relate primarily to the order of starlikeness and convexity of functions of the aforementioned classes.
LA - eng
KW - Complex harmonic functions; analytic conditions; convexity of order β; starlikeness of order β; complex harmonic functions; convexity of order ; starlikeness of order
UR - http://eudml.org/doc/268102
ER -

References

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  1. Adamczyk, G., Łazińska, A., On some generalization of coefficient conditions for complex harmonic mappings, Demonstratio Math. 38 (2) (2004), 317-326. Zbl1071.30004
  2. Avci, Y., Złotkiewicz E., On harmonic univalent mappings, Ann. Univ. Mariae Curie-Skłodowska Sec. A. 44 (1) (1990), 1-7. 
  3. Clunie, J., Sheil-Small, T., Harmonic univalent mappings, Ann. Acad. Sci. Fenn., Ser. A. I. Math., 9 (1984), 3-25. Zbl0506.30007
  4. Duren, P., Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004. Zbl1055.31001
  5. Ganczar, A., On harmonic univalent functions with small coefficients, Demonstratio Math. 34 (3) (2001), 549-558. Zbl0988.30009
  6. Jahangiri, J. M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235 (1999), 470-477. Zbl0940.30003
  7. Jakubowski, J. Z., Łazińska, A. and Sibelska, A., On some properties of complex harmonic mappings with a two-parameter coefficient condition, Math. Balkanica, New Ser. 18 (2004), 313-319. Zbl1094.30018
  8. Łazińska, A., On complex mappings in the unit disc with some coefficient conditions, Folia Sci. Univ. Techn. Resoviensis 199 (26) (2002), 107-116. Zbl1196.30011
  9. Mocanu, S. S., Miller, P. T., Differential Subordinations: Theory and Applications, Marcel Dekker, New York and Basel, 2000. Zbl0954.34003
  10. Pinchuk, B., Starlike and convex functions of order α, Duke Math. J. 35 (4) (1968), 721-734. Zbl0167.36101
  11. Robertson, M., On the theory of univalent functions, Ann. of Math. 37 (1936), 374-408. Zbl62.0373.05
  12. Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220 (1998), 283-289. Zbl0908.30013

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