On New Subclass of Harmonic Univalent Functions Associated with Modified q-Operator

Main Article Content

Shujaat Ali Shah, Asghar Ali Maitlo, Muhammad Afzal Soomro, Khalida Inayat Noor


In this article, we introduce new subclasses of harmonic univalent functions associated with the q-difference operator. The modified q-Srivastava-Attiya operator is defined and certain applications of this operator are discussed. We investigate the sufficient condition, distortion result, extreme points and invariance of convex combination of the elements of the subclasses.

Article Details


  1. O.P. Ahuja, J.M. Jahangiri, Noshiro-type harmonic univalent functions, Sci. Math. Jpn. 56 (2002), 1–7.
  2. G.E. Andrews, R. Askey, R. Roy, Special functions, encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, UK, 1999.
  3. Y. Avci, E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A 44 (1990), 1–7.
  4. N.E. Cho, T.H. Kim, Multiplier transformation and strongly close-to-convex functions, Bull. Korean Math. Soc. 40 (2003), 399–410.
  5. J. Clunie, T. Sheil-Small, Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 9 (1984), 3–25.
  6. M. Darus, K. Al-Shaqsi, On certain subclass of harmonic univalent functions, J. Anal. Appl. 6 (2008), 17–28.
  7. P.L. Duren, A survey of harmonic mappings in the plane, Texas Tech. Univ. Math. Ser. 18 (1992), 1–15.
  8. J. Dziok, Classes of harmonic functions associated with Ruscheweyh derivatives, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. (RACSAM) 113 (2019), 1315–1329
  9. J. Dziok, M. Darus, J. Sok´o l, T. Bulboaca, Generalizations of starlike harmonic functions, C. R. Math. Acad. Sci. Paris 354 (2016), 13–18.
  10. F.H. Jackson, On q-functions and a certain difference operator, Earth Environ. Sci. Trans. R. Soc. Edinburgh. 46 (1908), 253–281.
  11. J.M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math. Anal. Appl. 235 (1999), 470–477.
  12. J.M. Jahangiri, Harmonic univalent functions defined by q-calculus operators, Int. J. Math. Anal. Appl. 5 (2018), 39–43.
  13. J.M. Jahangiri, G. Murugusundaramoorthy, K. Vijaya, Starlikeness of Rucheweyh type harmonic univalent functions, J. Indian Acad. Math. 26 (2004), 191–200.
  14. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692.
  15. G. Murugusundramoorthy, J.M. Jahangiri, Ruscheweyh-type harmonic functions defined by q-differential operators, Khayyam J. Math. 5 (2019), 79–88.
  16. K.I. Noor, B. Malik, S.Z.H. Bukhari, Harmonic functions defined by a generalized fractional differential operator, J. Adv. Math. Stud. 2 (2009), 41–52.
  17. K.I. Noor, S. Riaz, M.A. Noor, On q-Bernardi integral operator, TWMS J. Pure Appl. Math. 8 (2017), 3–11.
  18. S.A. Shah, K.I. Noor, Study on q-analogue of certain family of linear operators, Turk. J. Math. 43 (2019), 2707–2714.
  19. T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc. 2 (1990), 237–248.
  20. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220 (1998), 283–289.
  21. H.M. Srivastava, A.A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integral Trans. Spec. Funct. 18 (2007), 207–216.
  22. S. Yal¸cın, A new class of Salagean-type harmonic univalent functions, Appl. Math. Lett. 18 (2005), 191–198.