On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator

Year 2018, Volume: 6 Issue: 2, 226 - 232, 15.10.2018

Sevtap Sümer Eker , Bilal Şeker

Abstract

In this paper, we introduce and investigate new subclasses of strongly bi-starlike and bi-starlike functions defined by Tremblay fractional derivative operator in the open unit disk. Also we obtain upper bounds for the coefficients $|a_{2}|$ and $|a_{3}|$ of functions belonging to these classes. Unlike recent studies, we use different technique for obtain the upper bounds on the coefficients $|a_{3}|$. Theorems proved in this paper generalizes the results given in [3].

Keywords

bi-starlike functions, Coefficient bounds and coefficient estimates, Fractional derivative, strongly bi-starlike functions

References

• [1] G. Akın and S. S¨umer Eker, Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative, C. R. Acad. Sci. S´er. I 352 (2014), 1005–1010.
• [2] R.M. Ali, S.K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Applied Mathematics Letters, 25 (2012) 344-351.
• [3] D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis and Its Applications, Kuwait; February 18-21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60. See also Studia Univ. Babes¸-Bolyai Math. 31 (2) (1986) 70-77.
• [4] P.L. Duren, Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
• [5] Z. Esa, A. Kilicman, R.W. Ibrahim, M. R.Ismail and S. K. S. Husain, Application of Modified Complex Tremblay Operator, AIP Conference Proceedings 1739, 020059 (2016); http://doi.org/10.1063/1.4952539.
• [6] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Applied Mathematics Letters, 24 (2011), 1569-1573.
• [7] R.W. Ibrahim, J.M. Jahangiri, Boundary fractional differential equation in a complex domain, Boundary Value Problems (2014) ; Article ID 66: 1 – 11.
• [8] S. S. Kumar, V. Kumar and V. Ravichandran, Estimates for the initial coefficients of bi-univalent functions, Tamsui Oxford J. Inform. Math. Sci. 29 (2013), 487–504.
• [9] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967) 63-68.
• [10] N. Magesh and J. Yamini, Coefficient bounds for a certain subclass of bi-univalent functions, Internat. Math. Forum 27 (2013), 1337–1344.
• [11] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978), 53-59.
• [12] S. Owa and H.M.Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987 ), 1057-1077.
• [13] Ch. Pommerenke, Univalent functions, G¨ottingen: Vandenhoeck Ruprecht 1975
• [14] W.Rudin, Real and Complex Analysis, McGraw-Hill Education ; 3 edition (May 1, 1986)
• [15] H. M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc. 23 (2015), 242–246.
• [16] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010) 1188-1192.
• [17] H.M. Srivastava and S. Owa, Some characterization and distortion theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Math.J. 106 (1987),1-28.
• [18] H.M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, Ellis Horwood Limited, Chichester and JohnWiley and Sons, NewYork, Chichester, Brisbane and Toronto, 1989.
• [19] H. M. Srivastava, S. S¨umer Eker and R. M. Ali, Coefficient Bounds for a certain class of analytic and bi-univalent functions, Filomat 29 (2015), 1839–1845.
• [20] H. M. Srivastava, S. S¨umer Eker, S. G. Hamidi, J. M.Jahangiri, Faber Polynomial Coefficient Estimates for Bi-univalent Functions Defined by the Tremblay Fractional Derivative Operator, Bulletin of the Iranian Mathematical Society (2018), 44 (1), 149–157.
• [21] S. S¨umer Eker, Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions, Turkish J Math (2016) 40: 641–646.
• [22] T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
• [23] R. Tremblay, Une Contribution ‘a la Th´eorie de la D´eriv´ee Fractionnaire, Ph.D. thesis, Laval University, Qu´ebec, 1974.