Coefficient estimates for a subclass of bi-univalent functions involving the q-derivatives operator

Shaba, Timilehin Gideon; Wanas, Abbas Kareem

doi:10.2478/gm-2021-0004

Abstract

In this current study, we introduce and investigate a new subclass of holormorphic and bi-univalent functions 𝕰 ^η,ϕ_q(ϑ) in the unit disk λ associated with q-derivative operator. The coefficient estimates |b₂| and |b₃| on the new subclass are obtained and important results are indicated.

References

[1] C. Abirami, N. Magesh, J. Yamini, Initial bounds for certain classes of biunivalent functions defined by Horadam polynomials, Abstr. Appl. Anal., Art. ID 7391058, 2020, 1-8.10.1155/2020/7391058
Search in Google Scholar Back to article
[2] A. Akgul, Finding initial coefficients for a class of bi-univalent functions given by q-derivative, AIP Conference Proceedings, vol. 1926, no. 1, 2018, 020001.10.1063/1.5020450
Search in Google Scholar Back to article
[3] A. Aral, V. G. Ravi, P. Agarwal, Applications of q-Calculus in Operator Theory, New York, NY : Springer, vol. 46, 1908, 253-281.
Search in Google Scholar Back to article
[4] I. Aldawish, T. Al-Hawary, B. A. Frasin, Subclasses of bi-univalent functions defined by Frasin differential operator, Mathematics, vol. 8, no. 5, 2020, https://doi.org/10.3390/math8050783.10.3390/math8050783
Search in Google Scholar Back to article
[5] H. Aldweby, M. Darus, Coefficient estimates of classes of Q-stalike and Q-convex functions, Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 26, no. 1, 2016, 21-26.
Search in Google Scholar Back to article
[6] H. Aldweby, M. Darus, A subclass of harmonic 𝔘 univalent functions associated with q-analogue of Dziok-Srivastava operator, ISRN Mathematical Analysis, Art. ID 382312, 2013, 1-6.10.1155/2013/382312
Search in Google Scholar Back to article
[7] H. Aldweby, M. Darus, Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent functions associated with q-derivative operator, Recent Trends in Pure and Applied Mathematics, vol. 2017, 2017, 1-9.
Search in Google Scholar Back to article
[8] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., vol. 25, no. 3, 2012, 344-351.10.1016/j.aml.2011.09.012
Search in Google Scholar Back to article
[9] S. Bulut, Certain subclasses of analytic and bi-univalent functions involving the q-derivative operator, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., vol. 66, no. 1, 2017, 108-114.10.1501/Commua1_0000000780
Search in Google Scholar Back to article
[10] D. A. Brannan, T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math., vol. 31, 1986, 70-77.
Search in Google Scholar Back to article
[11] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., vol. 2, no. 1, 2013, 49-60.10.7153/jca-02-05
Search in Google Scholar Back to article
[12] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
Search in Google Scholar Back to article
[13] B. A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat., vol. 43, no. 3, 2014, 383-389.
Search in Google Scholar Back to article
[14] B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., vol. 24, 2011, 1569-1573.10.1016/j.aml.2011.03.048
Search in Google Scholar Back to article
[15] S. P. Goyal, P. Goswami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. E-gyptian Math. Soc., vol. 20, no. 3, 2012, 179-182.10.1016/j.joems.2012.08.020
Search in Google Scholar Back to article
[16] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK. 35, 1990.
Search in Google Scholar Back to article
[17] F. H. Jackson, On q-definite integrals, The Quarterly Journal of Pure and Applied Mathematics, vol. 41, 1910, 193-203.
Search in Google Scholar Back to article
[18] F. H. Jackson, On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, vol. 46, 1908, 253-281.10.1017/S0080456800002751
Search in Google Scholar Back to article
[19] M. Lewin, On a coefficients problem of bi-univalent functions, Proc. Amer. Math. Soc., vol. 18, 1967, 63-68.10.1090/S0002-9939-1967-0206255-1
Search in Google Scholar Back to article
[20] N. Magesh, T. Rosy, S. Varma, Coefficient estimate problem for a new subclass of bi-univalent functions, J. Complex Anal., 2013, 1-3.10.1155/2013/474231
Search in Google Scholar Back to article
[21] G. Murugusundaramoorthy, N. Magesh, V. Prameela, Coefficient bounds for certain subclasses of bi-univalent functions, Abstr. Appl. Anal., Art. ID 573017, 2013, 1-3.10.1155/2013/573017
Search in Google Scholar Back to article
[22] G. Murugusundaramoorthy, C. Selvaraj, O. S. Badu, Coefficient estimates for pascu-type subclasses of bi-univalent function based on subordination, Int. J. of Nonlinear Sci., vol. 19, 2015, 47-52.
Search in Google Scholar Back to article
[23] U. H. Naik, A. B. Patil, On initial coefficient inequalities for certain new subclasses of bi-univalent functions, J. Egyptian Math. Soc., vol. 25, 2017, 291-293.10.1016/j.joems.2017.04.001
Search in Google Scholar Back to article
[24] S. Porwal, M. Darus, On a new subclass of bi-univalent functions, J. Egyptian Math. Soc., vol. 21, no. 3, 2013, 190-193.10.1016/j.joems.2013.02.007
Search in Google Scholar Back to article
[25] C. H. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, Göttingen, 1975.
Search in Google Scholar Back to article
[26] T. M. Seoudy, M. K. Aouf, Convolution properties for certain classes of analytic functions defined by q-derivative operator, Abstr. Appl. Anal., Art. ID 846719, 2014, 1-7.10.1155/2014/846719
Search in Google Scholar Back to article
[27] T. M. Seoudy, M. K. Aouf, Coefficient estimates of new classes of qstarlike and q-convex functions of complex order, J. Math. Inequal., vol. 10, no. 1, 2016, 135-145.10.7153/jmi-10-11
Search in Google Scholar Back to article
[28] H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., vol. 23, 2010, 1188-1192.10.1016/j.aml.2010.05.009
Search in Google Scholar Back to article
[29] H. M. Srivastava, A. Motamednezhad, E. A. Adegani, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, Art. ID 172, vol. 8, 2020, 1-12.10.3390/math8020172
Search in Google Scholar Back to article
[30] H. M. Srivastava, A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., vol. 59, no. 3, 2019, 493-503.
Search in Google Scholar Back to article
[31] A. K. Wanas, A. L. Alina, Applications of Horadam polynomials on Bazilevič bi-univalent function satisfying subordinate conditions, Journal of Physics: Conf. Series, vol. 1294, 2019, 1-6.10.1088/1742-6596/1294/3/032003
Search in Google Scholar Back to article
[32] Q. H. Xu, Y.C. Gui, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 2012, 11461-11465.10.1016/j.amc.2012.05.034
Search in Google Scholar Back to article

Coefficient estimates for a subclass of bi-univalent functions involving the q-derivatives operator

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