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

Abstract

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.

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

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

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].

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.
There are 23 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Sevtap Sümer Eker 0000-0002-2573-0726

Bilal Şeker

Publication Date October 15, 2018
Submission Date August 30, 2018
Acceptance Date October 11, 2018
Published in Issue Year 2018 Volume: 6 Issue: 2

Cite

APA Sümer Eker, S., & Şeker, B. (2018). On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator. Konuralp Journal of Mathematics, 6(2), 226-232.
AMA Sümer Eker S, Şeker B. On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator. Konuralp J. Math. October 2018;6(2):226-232.
Chicago Sümer Eker, Sevtap, and Bilal Şeker. “On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator”. Konuralp Journal of Mathematics 6, no. 2 (October 2018): 226-32.
EndNote Sümer Eker S, Şeker B (October 1, 2018) On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator. Konuralp Journal of Mathematics 6 2 226–232.
IEEE S. Sümer Eker and B. Şeker, “On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator”, Konuralp J. Math., vol. 6, no. 2, pp. 226–232, 2018.
ISNAD Sümer Eker, Sevtap - Şeker, Bilal. “On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator”. Konuralp Journal of Mathematics 6/2 (October 2018), 226-232.
JAMA Sümer Eker S, Şeker B. On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator. Konuralp J. Math. 2018;6:226–232.
MLA Sümer Eker, Sevtap and Bilal Şeker. “On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator”. Konuralp Journal of Mathematics, vol. 6, no. 2, 2018, pp. 226-32.
Vancouver Sümer Eker S, Şeker B. On Subclasses Of Bi-Starlike Functions Defined By Tremblay Fractional Derivative Operator. Konuralp J. Math. 2018;6(2):226-32.
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