Advertisement

On (p, q)-Quantum Calculus Involving Quasi-Subordination

  • S. Kavitha
  • Nak Eun Cho
  • G. MurugusundaramoorthyEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let (p, q) ∈ (0, 1). Let the function f be analytic in |z| < 1. Further, let the (p, q) be differential operator defined as \( {\displaystyle } {{\partial _{p,q}}}f \left ( z \right ) = \frac {{f\left ( pz \right ) - f\left ( {qz} \right )}}{{z\left ( {p - q} \right )}}, \quad |z|<1. \) In the current investigation, the authors apply the (p, q)-differential operator for few subclasses of univalent functions defined by quasi-subordination. Initial coefficient bounds for the defined new classes are obtained.

Notes

Acknowledgements

The work of the first author is supported by a grant from SDNB Vaishnav College for Women under Minor Research Project scheme. The work was completed when the first author was visiting VIT Vellore Campus for a research discussion with Prof. G.Murugusundaramoorthy during the second week of November 2017.

Conflicts of Interest The authors declare that they have no conflicts of interest regarding the publication of this paper.

References

  1. 1.
    R. M. Ali, V. Ravichandran, and N. Seenivasagan, Coefficient bounds for p-valent functions, Applied Mathematics and Computation, 187(1), 2007, 35–46.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam, The Fekete-Szego coefficient functional for transforms of analytic functions, Bulletin of the Iranian Mathematical Society, 35(2), 2009, 119–142.MathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Araci, U. Duran, M. Acikgoz and H. M. Srivastava, A certain (p, q)-derivative operator and associated divided differences, J. Inequal. Appl., (2016), 2016:301.MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A 24(13) (1991), L711–L718.MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Haji Mohd and M. Darus, Fekete-Szegő problems for quasi-subordination classes, Abstr. Appl. Anal. 2012, Art. ID 192956, 14 pp.Google Scholar
  6. 6.
    W. Ma and D. Minda,A unified treatment of some special classes of univalent functions, in Proceedings of the conference on complex Analysis, Z. Li, F. Ren, L. Lang and S. Zhang (Eds.), Int. Press (1994), 157–169.Google Scholar
  7. 7.
    M. Mursaleen, K. J. Ansari and A. Khan, Some approximation results by (p, q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput. 264 (2015), 392–402.MathSciNetzbMATHGoogle Scholar
  8. 8.
    F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proceedings of the American Mathematical Society, 20 (1969), 171–180.MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. Sahai and S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl. 335 (2007), 268–279.MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • S. Kavitha
    • 1
  • Nak Eun Cho
    • 2
  • G. Murugusundaramoorthy
    • 3
    Email author
  1. 1.Department of MathematicsS.D.N.B Vaishnav College for WomenChennaiIndia
  2. 2.Department of Applied MathematicsPukyong National UniversityBusanRepublic of Korea
  3. 3.Department of Mathematics, School of Advanced SciencesVITVelloreIndia

Personalised recommendations