Certain Subclasses of Bi-Univalent Functions Defined by $q$-Analogue of Ruscheweyh Differential Operator
Abstract
In this paper, we find a new subclasses of the function class $\sum$ of bi-univalent functions defined in the open unit disk, which are associated with the $q$-analogue of Ruscheweyh differential operator and satisfy some subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|v_2|$ and $|v_3|$ for functions in the new subclasses introduced here.
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References
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\bibitem{rst} Ruscheweyh, S., {\it New criteria for univalent functions}, Proc. Am. Math. Soc., 49, 109--115, (1975).
\bibitem{hmsakmpg} Srivastava, H. M., Mishra, A. K. and Gochhayat, P., {\it Certain subclasses of analytic and bi-univalent functions}, Appl. Math. Lett., 23(10), 1188--1192, (2010).
\bibitem{thso} Hayami, T. and Owa, S., {\it Coefficient bounds for bi-univalent functions}, Panam. Math. J., 22(4), 15--26, (2012).
\bibitem{qhxhmszl} Xu. Q-H., Srivastava, H. M. and Li, Z., {\it A certain subclass of analytic and close-to-convex functions}, Appl. Math. Lett., 24(3), 396--401, (2011).
\bibitem{xflapw} Li, X-F. and Wang, A-P., {\it Two new subclasses of bi-univalent functions}, Int. Math. Forum, 7, 1495--1504, (2012).
\bibitem{dabjcwek} Brannan, D. A. and Clunie, J. and Kirwan, W. E., {\it Coefficient estimates for a class of star-like functions}, Can. J. Math., 22, 476--485, (1970).
\bibitem{dabtst} Brannan, D. A. and Taha, T. S., {\it On some classes of bi-univalent functions}, Stud. Univ. Babe{\textcommabelow{s}}-Bolyai, Math., 31(2), 70--77, (1986).
\bibitem{ed} Deniz, E., {\it Certain subclasess of bi-univalent functions satisfying subordinate conditions}, Journal of Classical Analysis, 2(1), 49--60, (2013).
\bibitem{jfh} Jackson, F. H., {\it XI.—On $q$-Functions and a certain Difference Operator}, Transactions of the Royal Society of Edinburgh, 46(2), 253--281, (1909).
\bibitem{tjlscmi} Latha, T. J. and Indrani, S. C. M., {\it Coefficient Estimates for Bi-univalent Ma-Minda Starlike and Convex Functions}, Journal of Emerging Technologies and Innovative Research, 6(6), 68--74, (2019).
\bibitem{ksrd} Kanas, S. and R{\u{a}}ducanu, D., {\it Some class of analytic functions related to conic domains}, Math. Slovaca, 64(5), 1183--1196, (2014).
\bibitem{ml} Lewin, M., {\it On a coefficient problem for bi-univalent functions}, Proc. Am. Math. Soc., 18, 63--68, (1967).
\bibitem{wcmdm} Ma, W. and Minda, D., {\it A unified treatment of some special classes of univalent functions}, Proceedings of the conference on complex analysis, held June 19-23, 1992 at the Nankai Institute of Mathematics, Tianjin, China, Cambridge, MA: International Press, Pages 157--169, (1994).
\bibitem{rst} Ruscheweyh, S., {\it New criteria for univalent functions}, Proc. Am. Math. Soc., 49, 109--115, (1975).
\bibitem{hmsakmpg} Srivastava, H. M., Mishra, A. K. and Gochhayat, P., {\it Certain subclasses of analytic and bi-univalent functions}, Appl. Math. Lett., 23(10), 1188--1192, (2010).
\bibitem{thso} Hayami, T. and Owa, S., {\it Coefficient bounds for bi-univalent functions}, Panam. Math. J., 22(4), 15--26, (2012).
\bibitem{qhxhmszl} Xu. Q-H., Srivastava, H. M. and Li, Z., {\it A certain subclass of analytic and close-to-convex functions}, Appl. Math. Lett., 24(3), 396--401, (2011).
\bibitem{xflapw} Li, X-F. and Wang, A-P., {\it Two new subclasses of bi-univalent functions}, Int. Math. Forum, 7, 1495--1504, (2012).
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2025-08-24
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