On subclasses of analytic functions defined by using Tremblay fractional derivative operator

[1] M.K. Aouf, R.M. El-Ashwah and S.M. El-Deeb, Subordination results for certain subclasses of uniformly starlike and convex functions defined by convolution, European J. Pure Appl. Math. 3 (2010), no. 5, 903–917.
[2] M.K. Aouf, R.M. El-Ashwah, A.A.M. Hassan and A.H. Hassan, On subordination results for certain new classes of analytic functions defined by using Salagean operator, Bull. Math. Anal. Appl. 4 (2012), no. 1, 239–246.
[3] A.A. Attiya, On some application of a subordination theorems, J. Math. Anal. Appl. 311 (2005), 489–494.

[4] T. Bulboaca, Differential Subordinations and Superordinations, Recent Results, House of Science Book Publ. Cluj-Napoca, 2005.
[5] Z. Esa, A. Kilicman, R.W. Ibrahim, M.R. Ismail and S.K.S. Husain, Application of modified complex Tremblay operator, AIP Conf. Proc., 2016, 020059.
[6] B.A. Frasin, Subordination results for a class of analytic functions defined by a linear operator, J. Inequal. Pure Appl. Math. 7 (2006), no. 4, Art. 134, 1–7.
[7] A.W. Goodman, Univalent functions and analytic curves, Proc. Amer. Math. Soc. 8 (1957), no. 3, 598-601.
[8] V.P. Gupta and P.K. Jain, Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc. 14 (1976), 409–416.
[9] R.W. Ibrahim and J.M. Jahangiri, Boundary fractional differential equation in a complex domain, Boundary Value Prob. 2014 (2014), Article ID 66.
[10] T.H. MacGregor, The radius of convexity for starlike function of order α, Proc. Amer. Math. Sec. 14 (1963), 71–76.
[11] S.S. Miller and P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978), 289–305.
[12] B. Pinchuk, On the starlike and convex functions of order α, Duke Math. J. 35 (1968), 721–734.
[13] A. Schild, On starlike function of order α, Amer. J. Math. 87 (1965), 65–70.
[14] A. Schild and H. Silverman, Convolutions of univalent functions with negative coefficients, Ann. Univ. Mariae Curie-Sk lodowska Sect. A-Math. 29 (1975), 99–107.

[15] H.M. Srivastava and M.K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients, J. Math. Anal. Appl. 171 (1992), 1–13.
[16] H.M. Srivastava and M.K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients II, J. Math. Anal. Appl. 192 (1995), 673–688.
[17] H.M. Srivastava and A.A. Attiya, Some subordination results associated with certain subclass of analytic functions, J. Inequal. Pure Appl. Math. 5 (2004), no. 4, Art. 82, 1–6.
[18] H.M. Srivastava and S.S. Eker, Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett. 21 (2008), 394–399.
[19] H.M. Srivastava, S.S. Eker, S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc. 44 (2018), no. 1, 149–157.

[20] H.M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, Horwood Limited, Chichester and John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
[21] H.M. Srivastava and S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math. J. 106 (1987), 1–28.
[22] R. Tremblay, Une Contribution a la Th´eorie de la D´eriv´ee Fractionnaire, Ph.D. thesis, Laval University, Qu´ebec, 1974.
[23] H.S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc. 12 (1961),689–693.