Spirallikeness properties on Salagean-type harmonic univalent functions

Document Type : Research Paper


Department of Mathematics, Payme Noor University, P. O. Box 19395-4697 Tehran, IRAN.


Abstract. In this paper, we define and investigate a new class of spirallike harmonic functions defined by a Salagean differential operator and we obtain a coefficient inequality for the functions in this class. Following, we investigated convolution and obtain the order of convolution consistence for certain spirallike harmonic univalent functions with negative coefficients.


[1] O. P. Ahuja, J. M. Jahangiri and H. Silverman, Convolutions for special classes of harmonic univalent functions, Appl. Math. Lett. 16(2003), 905-909.
[2] Ahuja, O. P, Recent advances in the theory of harmonic univalent mappings in the plane , Math. Student, 83(2014), 125154.
[3] Ahuja, O. P., Planar harmonic univalent and related mappings, J. Inequal. Pure Appl. Math., 6 (2005), 1-19.
[4] Om P. Ahuja, Sumit Nagpal and V. Ravichandran A technique of constructing planar harmonic mappings and their properties, Kodai Math. J. 40 (2017), 278288.
[5] I.C.M. Balaetei, An integral operator associated with di erential super-ordinations, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17(2009), 37-44.
[6] U. Bednarz and J. Sokol, On order of convolution consistence of the analytic functions, Stud. Univ. Babes-Bolyai Math., 55(2010), 41-50.
[7] D. Bshoutya, S. S. Joshib and S.B. Joshi, On close-to-convex harmonic mappings, Complex Variables and Elliptic Equations, 58(2013), 11951199
[8] J. Clunie and T. Sheil, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I Math. 9(1984), 3-25.
[9] P. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156. Cambridge Univ. Press, Cambridge, 2004.
[10] J. M. Jahangiri, Harmonic functions starlike in unite dist, J. Math. Anal. Appl., 235(1999), 470-477.
[11] R. J. Libera, Univalent  -spiral functions, Canad. J. Math. 19 (1967), 449-456.
[12] S. Muir, Weak subordination for convex univalent harmonic functions, J. Math. Anal. Appl., 348(2008), 862871.
[13] S. Ponnusamy and V. Singh, Convolution properties of some classes analytic functions, Za- piski Nauchnych Seminarov POMI 226(1996), 138-154.
[14] L. Liulan, S. Ponnusamy, Disk of convexity of sections of univalent harmonic functions, J. Math. Anal. Appl., 408(2013), 408, 589596.
[15] St. Ruscheweyh and T. Sheil-Small, Hadamard products of schlicht functions and the Plya- Schoenberg conjecture, Comment. Math. Helv. 48(1973), 119-135
[16] G. S. Salagean: On some classes of univalent functions, Seminar of geometric function theory, Cluj - Napoca, 1983.
[17] T. Sheil-Small, Constants for planar harmonic mapping, J. London Math. Soc. 2(1990), 237-248.
[18] H. Silverman, Harmonic univalent functions with negative coecients, J. Math. Anal. Appl., 220(1998), 283-289.
[19] Wang, Z-G., Liu, Z-H., Li, Y-C., On the linear combinations of harmonic univalent mappings, J. Math. Anal. Appl., 400( 2013), 452459.
  • Receive Date: 12 February 2021
  • Revise Date: 14 December 2021
  • Accept Date: 16 December 2021
  • First Publish Date: 17 December 2021