On subclasses of bi-convex functions defined by Tremblay fractional derivative operator
DOI:
https://doi.org/10.24193/subbmath.2019.4.02Keywords:
Bi-convex functions, Tremblay fractional derivative, coefficient bounds and coefficient estimates.Abstract
We introduce and investigate new subclasses of analytic and bi-univalent functions defined by modified Tremblay operator in the open unit disk. Also we obtain upper bounds for the coefficients of functions belonging to these classes.
Mathematics Subject Classification (2010): 30C45, 30C50, 30C80.
References
Akın, G., Su¨mer Eker, S., Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative, C.R. Math. Acad. Sci. Paris, 352(2014), 1005-1010.
Ali, R.M., Lee, S.K., Ravichandran, V., Supramaniam, S., Coefficient estimates for bi- univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25(2012), 344-351.
Caglar, M., Deniz, E., Srivastava, H.M., Second Hankel determinant for certain subclasses of bi-univalent functions, Turkish J. Math., 41(2017), 694–706.
Deniz, E., Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2(2013), 49-60.
Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, Stud. Univ. Babe¸s-Bolyai Math., 31(1986), no. 2, 70-77.
Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S.M. Mazhar, 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.
Duren, P.L., Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
Esa, Z., Kilicman, A., Ibrahim, R.W., Ismail M.R., Husain, S.K.S., Application of Modified Complex Tremblay Operator, AIP Conference Proceedings 1739, 020059 (2016), http://doi.org/10.1063/1.4952539.
Ibrahim, R.W., Jahangiri, J.M., Boundary fractional differential equation in a complex domain, Bound. Value Probl., 2014, Art. ID 66: 1-11.
Kumar, S.S., Kumar V., Ravichandran, V., Estimates for the initial coefficients of bi- univalent functions, Tamsui Oxf. J. Inf. Math. Sci., 29(2013), 487-504.
Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18(1967), 63-68.
Magesh, N., Yamini, J., Coefficient bounds for a certain subclass of bi-univalent functions, Internat. Math. Forum, 27(2013), 1337-1344.
Owa, S., On the distortion theorems I, Kyungpook Math. J., 18(1978), 53-59.
Owa, S., Srivastava, H.M., Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39(1987), 1057-1077.
Pommerenke, Ch., Univalent Functions G¨ottingen, Vandenhoeck Ruprecht, 1975. [16] Rudin, W., Real and Complex Analysis, McGraw-Hill Education, 3rd edition, 1986.
Srivastava, H.M., Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions, in: Nonlinear Analysis: Stability; Approximation; and Inequalities (Panos M. Pardalos,Pando G. Georgiev and Hari M. Srivastava, Eds.), Springer Series on Optimization and Its Applications, Vol. 68, Springer-Verlag, Berlin, Heidelberg and New York, 2012, 607-630.
Srivastava, H.M., Bansal, D., Coefficient estimates for a subclass of analytic and bi- univalent functions, J. Egyptian Math. Soc., 23(2015), 242-246.
Srivastava, H.M., Mishra, A.K., Gochhayat, P., Certain subclasses of analytic and bi- univalent functions, Appl. Math. Lett., 23(2010), 1188-1192.
Srivastava, H.M., Owa, S., Some characterization and distortion theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Math. J., 106(1987), 1-28.
Srivastava, H.M., Owa, S., Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, Ellis Horwood Limited, Chichester and John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
Srivastava, H.M., Su¨mer Eker, S., Ali, R.M., Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(2015), 1839-1845.
Su¨mer Eker, S., Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions, Turkish J. Math., 40(2016), 641-646.
Su¨mer Eker, S., Coefficient estimates for new subclasses of m-fold symmetric bi-univalent functions, Theory Appl. Math. Comput. Sci., 6(2)(2016), 103-109.
Taha, T.S., Topics in Univalent Function Theory, Ph. D. Thesis, University of London, 1981.
Tremblay, R., Une Contribution ‘a la Th´eorie de la D´eriv´ee Fractionnaire, Ph. D. Thesis, Laval University, Qu´ebec, 1974.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2019 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
