A class of harmonic univalent functions defined by the q-derivative operator

Document Type : Research Paper


Department of Mathematics, College of Education for Pure Sciences, University of Anbar, Ramadi, Iraq


In this paper, a class of harmonic univalent functions has been studied by using q-analogue of the derivative operator for complex harmonic functions. We have obtained a sufficient condition, a representation theorem for this harmonic univalent functions class and some other geometric properties.


[1] I. Aldawish and M. Darus, Starlikeness of q-differential operator involving quantum calculus, Korean J. Math.
22(4) (2014) 699–709.
[2] K. Al-Shaqsi and M. Darus, On univalent functions with respect to k-symmetric points defined by a generalized
Ruscheweyh derivatives operator, J. Anal. Appl. 7(1) (2009) 53–61.
[3] S. Elhaddad, H. Aldweby and M. Darus, Some properties on a class of harmonic univalent functions defined by
q-analogue of Ruscheweyh operator, J. Math. Anal. 9(4) (2018) 28–35.[4] D.O. Jackson, T. Fukuda, O. Dunn and E. Majors, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910)
[5] J.M. Jahangiri, Harmonic univalent functions defined by q-calculus operators, arXiv preprint
[6] A.R.S. Juma and L.I. Cotˆırla, On harmonic univalent function defined by generalized salagean derivatives, Acta
Universitatis Apulensis 23 (2010) 179–188.
[7] S. Khan, S. Hussain, M.A. Zaighum and M. Darus, A subclass of uniformly convex functions and a corresponding
subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator, Math. Slovaca
69(4) (2019) 825–832.
[8] O. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42(10)
(1936) 689–692.
[9] O. Mishra and J. Sok´o l, Generalized q-starlike harmonic functions, Revista de la Real Academia de Ciencias
Exactas, F´ıs. Naturales Serie A. Mat. 115(4) (2021) 1–14.
[10] A.O. Mostafa and M.K. Aouf, Harmonic subclass of univalent functions defined by modified q-difference operator,
Afrika Mat. 32 (2021) 1323—1331.
[11] S. A. Porwal and A. N. Gupta, An application of q-calculus to harmonic univalent functions, J. Quality Measure.
Anal. 14(1) (2018) 81–90.
[12] M.S.U. Rehman, Q.Z. Ahmad, H.M. Srivastava, N. Khan, M. Darus and B. Khan, Applications of higher-order
q-derivatives to the subclass of q-starlike, AIMS Math. 6(2) (2021) 1110–1125.
[13] H.M. Srivastava, N. Khan, S. Khan, Q.Z. Ahmad and B. Khan, A class of k-symmetric harmonic functions
involving a certain q-derivative operator, Math. 9(15) (2021) 1812.
[14] H.M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric
function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci. 44(1) (2020) 327–344.
Volume 13, Issue 1
March 2022
Pages 2713-2722
  • Receive Date: 16 October 2021
  • Revise Date: 07 November 2021
  • Accept Date: 11 December 2021
  • First Publish Date: 27 December 2021