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.
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R. M. Ali, V. Ravichandran, and N. Seenivasagan, Coefficient bounds for p-valent functions, Applied Mathematics and Computation, 187(1), 2007, 35–46.
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.
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.
R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A 24(13) (1991), L711–L718.
M. Haji Mohd and M. Darus, Fekete-Szegő problems for quasi-subordination classes, Abstr. Appl. Anal. 2012, Art. ID 192956, 14 pp.
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.
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.
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.
V. Sahai and S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl. 335 (2007), 268–279.
M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9.
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.
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Kavitha, S., Cho, N.E., Murugusundaramoorthy, G. (2018). On (p, q)-Quantum Calculus Involving Quasi-Subordination. In: Madhu, V., Manimaran, A., Easwaramoorthy, D., Kalpanapriya, D., Mubashir Unnissa, M. (eds) Advances in Algebra and Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01120-8_25
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DOI: https://doi.org/10.1007/978-3-030-01120-8_25
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