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Coefficient bounds for a certain subclass of analytic and bi-univalent functions

Year 2019, Volume: 68 Issue: 2, 1492 - 1505, 01.08.2019
https://doi.org/10.31801/cfsuasmas.540642

Abstract

In this paper, we introduce and investigate a new subclass of the a-nalytic and bi-univalent functions in the open unit disk in the complex plane. For the functions belonging to this class, we obtain estimates on the first three coefficients in their Taylor-Maclaurin series expansion. Some interesting corollaries and applications of the results obtained here are also discussed.

References

  • Ali R. M., Lee S. K., Ravichandran V. and Supramanian S., Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25(3), (2012), 334--351.
  • Brannan D. A. and Clunie J. G., Aspects of Contemporary Complex Analysis, Proceeding of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1-20, 1979, (Academic Press, New York and London, 1980).
  • Brannan D. A. and Taha T. S., On some classes of bi-univalent functions, Mathematical Analysis and Its Applications 3, (1985), 18-21.
  • Bulut S., Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad J. Mat. 43(2), 59--65 , 2013.
  • Çağlar M., Orhan H. and Yağmur N., Coefficient bounds for new subclasses of bi-univalent functions, Filomat 27(7), 1165--1171, 2013.
  • Deniz E., Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2(1), (2013), 49--60.
  • Frasin B. A. and Aouf M. K., New subclasses of bi-univalent functions, Appl. Math. Lett. 24(9), (2011), 1569--1573.
  • Goyal S. P. and Goswami P., Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc. 20, (2012), 179--182.
  • Hayami T. and Owa S., Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22(4), 15-26, 2012.
  • Kim Y. C. and Srivastava H. M., Some subordination properties for spirallike functions, Appl. Math. Comput. 203(2), (2008), 838--842.
  • Lewin M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18, (1967), 63--68.
  • Ma W. C. and Minda D., A unified treatment of some special classes of functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, , (1992),157-169, Conf. Proc. Lecture Notes anal. I, Int. Press, Cambridge, MA, 1994.
  • Magesh N. and Yamini J., Coeffcient bounds for certain subclasses of bi-univalent functions, Internat. Math. Forum 8(27), (2013), 1337--1344.
  • Miller S. S. and Mocanu P. T., Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics 225, Dekker, New York, 2000.
  • Murugusundaramoorthy G., Magesh N. and Prameela V., Coefficient bounds for certain subclasses of bi-univalent functions, Abs. Appl. Anal., (2013), Article Id 573017, 3 pages.
  • Netanyahu E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1, Arch. Rational Mech. Anal. 32, (1969), 100--112.
  • Orhan H., Magesh N. and Balaji V. K., Initial Coefficient Bounds for a General Class of Bi-Univalent Functions, Filomat 29(6), (2015), 1259--1267.
  • Pommerenke C., Univalent Functions, Vandenhoeck and Rupercht, Göttingen, 1975.
  • Prema S. and Keerthi B. S., Coefficient bounds for certain subclasses of analytic functions, J. Math. Anal. 4(1), (2013), 22-27.
  • Ravichandran V., Polatoglu Y., Bolcal M. and Sen A., Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34, (2005), 9--15.
  • Sivaprasad Kumar S., Kumar V. and Ravichandran V., Estimates for the initial coefficients of bi-univalent functions, (2012), arXiv:1203.5480v1.
  • Srivastava H. M., Mishra A. K. and Gochhayat P., Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23, (2010), 1188--1192.
  • 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, Editors), Springer Series on Optimization and Its Applications Vol. 68, Springer-Verlag, Berlin, Heidelberg and New York, (2012), 607--630.
  • Srivastava H. M., Bulut S., Çağlar M. and Yağmur N., Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27(5), (2013), 831--842.
  • Srivastava H. M., Murugusundaramoorthy G. and Magesh N., On certain subclasses of bi-univalent functions associated with Hohlov operator, Global J. Math. Anal. 1(2), (2013), 67--73.
  • Srivastava H. M., Xu Q. H. and Wu G. P., Coefficient estimates for certain subclasses of spiral-like functions of complex order, Appl. Math. Lett. 23(7), (2010), 763--768.
  • Xu Q. H., Gui Y. C. and Srivastava H. M., Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25(6), (2012), 990-994.
  • Xu Q. H., Xiao H. G. and Srivastava H. M., A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218(23), (2012), 11461--11465.
  • Xu Q. H., Cai Q. M. and Srivastava H. M., Sharp coefficient estimates for certain subclasses of starlike functions of complex order, Appl. Math. Comput. 225, (2013), 43--49.
Year 2019, Volume: 68 Issue: 2, 1492 - 1505, 01.08.2019
https://doi.org/10.31801/cfsuasmas.540642

Abstract

References

  • Ali R. M., Lee S. K., Ravichandran V. and Supramanian S., Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25(3), (2012), 334--351.
  • Brannan D. A. and Clunie J. G., Aspects of Contemporary Complex Analysis, Proceeding of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1-20, 1979, (Academic Press, New York and London, 1980).
  • Brannan D. A. and Taha T. S., On some classes of bi-univalent functions, Mathematical Analysis and Its Applications 3, (1985), 18-21.
  • Bulut S., Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad J. Mat. 43(2), 59--65 , 2013.
  • Çağlar M., Orhan H. and Yağmur N., Coefficient bounds for new subclasses of bi-univalent functions, Filomat 27(7), 1165--1171, 2013.
  • Deniz E., Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2(1), (2013), 49--60.
  • Frasin B. A. and Aouf M. K., New subclasses of bi-univalent functions, Appl. Math. Lett. 24(9), (2011), 1569--1573.
  • Goyal S. P. and Goswami P., Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc. 20, (2012), 179--182.
  • Hayami T. and Owa S., Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22(4), 15-26, 2012.
  • Kim Y. C. and Srivastava H. M., Some subordination properties for spirallike functions, Appl. Math. Comput. 203(2), (2008), 838--842.
  • Lewin M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18, (1967), 63--68.
  • Ma W. C. and Minda D., A unified treatment of some special classes of functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, , (1992),157-169, Conf. Proc. Lecture Notes anal. I, Int. Press, Cambridge, MA, 1994.
  • Magesh N. and Yamini J., Coeffcient bounds for certain subclasses of bi-univalent functions, Internat. Math. Forum 8(27), (2013), 1337--1344.
  • Miller S. S. and Mocanu P. T., Differential subordinations, Monographs and Textbooks in Pure and Applied Mathematics 225, Dekker, New York, 2000.
  • Murugusundaramoorthy G., Magesh N. and Prameela V., Coefficient bounds for certain subclasses of bi-univalent functions, Abs. Appl. Anal., (2013), Article Id 573017, 3 pages.
  • Netanyahu E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1, Arch. Rational Mech. Anal. 32, (1969), 100--112.
  • Orhan H., Magesh N. and Balaji V. K., Initial Coefficient Bounds for a General Class of Bi-Univalent Functions, Filomat 29(6), (2015), 1259--1267.
  • Pommerenke C., Univalent Functions, Vandenhoeck and Rupercht, Göttingen, 1975.
  • Prema S. and Keerthi B. S., Coefficient bounds for certain subclasses of analytic functions, J. Math. Anal. 4(1), (2013), 22-27.
  • Ravichandran V., Polatoglu Y., Bolcal M. and Sen A., Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34, (2005), 9--15.
  • Sivaprasad Kumar S., Kumar V. and Ravichandran V., Estimates for the initial coefficients of bi-univalent functions, (2012), arXiv:1203.5480v1.
  • Srivastava H. M., Mishra A. K. and Gochhayat P., Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23, (2010), 1188--1192.
  • 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, Editors), Springer Series on Optimization and Its Applications Vol. 68, Springer-Verlag, Berlin, Heidelberg and New York, (2012), 607--630.
  • Srivastava H. M., Bulut S., Çağlar M. and Yağmur N., Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27(5), (2013), 831--842.
  • Srivastava H. M., Murugusundaramoorthy G. and Magesh N., On certain subclasses of bi-univalent functions associated with Hohlov operator, Global J. Math. Anal. 1(2), (2013), 67--73.
  • Srivastava H. M., Xu Q. H. and Wu G. P., Coefficient estimates for certain subclasses of spiral-like functions of complex order, Appl. Math. Lett. 23(7), (2010), 763--768.
  • Xu Q. H., Gui Y. C. and Srivastava H. M., Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25(6), (2012), 990-994.
  • Xu Q. H., Xiao H. G. and Srivastava H. M., A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218(23), (2012), 11461--11465.
  • Xu Q. H., Cai Q. M. and Srivastava H. M., Sharp coefficient estimates for certain subclasses of starlike functions of complex order, Appl. Math. Comput. 225, (2013), 43--49.
There are 29 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Nizami Mustafa 0000-0001-9640-8526

Veysel Nezir 0000-0002-2758-0274

Publication Date August 1, 2019
Submission Date February 28, 2018
Acceptance Date November 8, 2018
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Mustafa, N., & Nezir, V. (2019). Coefficient bounds for a certain subclass of analytic and bi-univalent functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1492-1505. https://doi.org/10.31801/cfsuasmas.540642
AMA Mustafa N, Nezir V. Coefficient bounds for a certain subclass of analytic and bi-univalent functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):1492-1505. doi:10.31801/cfsuasmas.540642
Chicago Mustafa, Nizami, and Veysel Nezir. “Coefficient Bounds for a Certain Subclass of Analytic and Bi-Univalent Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 1492-1505. https://doi.org/10.31801/cfsuasmas.540642.
EndNote Mustafa N, Nezir V (August 1, 2019) Coefficient bounds for a certain subclass of analytic and bi-univalent functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 1492–1505.
IEEE N. Mustafa and V. Nezir, “Coefficient bounds for a certain subclass of analytic and bi-univalent functions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 1492–1505, 2019, doi: 10.31801/cfsuasmas.540642.
ISNAD Mustafa, Nizami - Nezir, Veysel. “Coefficient Bounds for a Certain Subclass of Analytic and Bi-Univalent Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 1492-1505. https://doi.org/10.31801/cfsuasmas.540642.
JAMA Mustafa N, Nezir V. Coefficient bounds for a certain subclass of analytic and bi-univalent functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1492–1505.
MLA Mustafa, Nizami and Veysel Nezir. “Coefficient Bounds for a Certain Subclass of Analytic and Bi-Univalent Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 1492-05, doi:10.31801/cfsuasmas.540642.
Vancouver Mustafa N, Nezir V. Coefficient bounds for a certain subclass of analytic and bi-univalent functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):1492-505.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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