Coefficient estimates and Fekete- Szegὅ inequality for a subclass of Bi-univalent functions defined by symmetric Q-derivative operator by using Faber polynomial techniques

G. Saravanan, K. Muthunagai

Abstract


In this article we have defined a subclass of Bi-univalent functions using symmetric q- derivative operator and estimated the bounds for the coefficients using Faber polynomial techniques. We also have obtained the bounds for the linear functional which is popularly known as Fekete- Szegὅ problem.

Full Text:

PDF

References


H. Airault, “Symmetric sums associated to the factorization of Grunsky coefficients,” In Conference, Groups and symmetries, Montreal, Canada, pp. 27-29, 2007.

H. Airault, “Remarks on Faber polynomials”, Int. Math. Forum, vol. 3, pp. 9-12, 2008.

H. Airault and A. Bouali, “Differential Calculus on the Faber polynomials,” Bull. Sci. Math., vol. 130(3), pp. 179-222, 2006.

H. Airault and J. Ren, “An algebra of differential operators and generalized functions on the set of univalent function,” Bull. Sci. Math., vol. 126(5), pp. 343-367, 2002.

H. Aldweby and M. A. Darus, “Subclass of harmonic univalent functions associated with q-analogue of Dziok- Srivastava operator,” ISRN Math. Anal., 2013(2013), Article ID 382312, 6 pages.

Altinkaya, and S. Yalçn, “The Fekete- Szegὅ problem for a general class of bi-univalent functions satisfying subordinate conditions,” Sahand Communications in Mathematical Analysis(SCMA), vol. 5(1), pp. 1-7, 2017.

Altinkaya, and S. Yalçn, “Faber polynomial coefficient estimates for a class of bi- univalent functions based on the symmetric q-derivative operator,” Journal of Fractional Calculus and Applications, vol. 8(2), pp. 79-87, 2017.

M. Aydoğan , Y. Kahramaner and Y. Polatoğlu, “Close-to-Convex Functions Defined by Fractional Operator,” Appl. Math. Sci., vol. 7, pp. 2769-2775, 2013.

K. L. Brahim and Y. Sidomou, “On some symmetric q-special functions,” Le Mat., vol. 68, pp. 107-122, 2013.

D. A. Brannan and J. G. Clunie, “Aspects of Contemporary Complex Analysis,” Academic Press, London and New York, 1980.

R. M. El-Ashwah, “Subclasses of bi-univalent functions defined by convolution,” Journal of the Egyptian Mathematical Society, vol. 22(3), pp. 348-351, 2014.

G. Faber, “Uber polynomische Entwickelungen,” Math. Ann. vol 57(3), pp. 389-408, 1903.

B. A. Frasin, and M. K. Aouf, “New subclasses of bi-univalent functions,” Applied Mathematics Letters, vol. 24, pp. 1569-1573, 2011.

G. Gasper, and M. Rahman, “Basic Hypergeometric Series,” Cambridge Univ. Press, Cambridge, MA, 1990.

S. G. Hamidi, and M. Jahangiri, “Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations,” Bull. Iranian Math. Soc., vol. 41(5), pp. 1103-1119, 2015.

T. Hayami and S. Owa,“Coefficient bounds for bi-univalent functions,” Panamerican Mathematical Journal, vol. 22(4), pp. 15-26, 2012.

F. H. Jackson, “On q-functions and a certain difference operator,” Trans. Royal Soc. Edinburgh, vol. 46, pp. 253-281, 1908.

M. Lewin, “On a Coefficient problem for bi-univalent functions,” Proc. Amer. Math.Soc., vol. 18, pp. 63-68, 1967.

A. Mohammed, and M. A. Darus, “generalized operator involving the q-hypergeometric function” Mat. Vesnik., vol. 65, pp. 454-465, 2013.

H. E. Ozkan Ucar, “Cofficient inequalties for q-starlike functions,” Appl. Math. Comp., vol. 276, pp. 122-126, 2016.

Y. Polatoglu, “Growth and distortion theorems for generalized q-starlike functions,” Adv. Math., Sci. J., vol. 5, pp. 7-12, 2016.

C. Pommerenke, “Univalent Functions,” Vandenhoeck & Ruprecht, Gottingen, 1975.

S. D. Purohit and R. K. Raina, “Fractional q -calculus and certain subclass of univalent analytic functions,” Mathematica, vol. 55, pp. 62-74, 2013.

G. Saravanan and K. Muthunagai, “Co-efficient estimates for the class of Bi-Quasi-Convex functions using Faber polynomials,” Far East J.Math. Sci. (FJMS), vol. 102(10), pp. 2267-2276. 2017.

R. Vijaya, T. V. Sudharsan and S. Sivasubramanian, “Coefficient Estimates for Certain Subclasses of Biunivalent Functions Defined by Convolution,” International Journal of Analysis, Article ID 6958098,(2016),5 pages.

Q. H. Xu, Y. C. Gui, and H. M. Srivastava, “Coefficient estimates for a certain subclass of analytic and bi-univalent functions,” Applied Mathematics Letters, vol. 25(6), pp. 990-994, 2012.




DOI: http://dx.doi.org/10.21533/pen.v6i1.285

Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 Saravanan G

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

ISSN: 2303-4521

Digital Object Identifier DOI: 10.21533/pen

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License