Tbilisi Mathematical Journal

Certain classes of bi-univalent functions with bounded boundary variation

H. Orhan, N. Magesh, and V. K. Balaji

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In their pioneering work dated 2010 on the subject of bi-univalent functions, Srivastava et al. actually revived the study of the coefficient problems involving bi-univalent functions in recent years. Inspired by the pioneering work of Srivastava et al., there has been triggering interest to study the coefficient problems for many different subclasses of bi-univalent functions. Motivated largely by a number of sequels to the investigation by Srivastava et al., in this paper, we consider certain classes of bi-univalent functions to obtain the estimates of their second and third Taylor-Maclaurin coefficients. Further, certain special cases are also indicated. Some interesting remarks about the results presented here are also discussed.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 4 (2017), 17-27.

Dates
Received: 31 January 2017
Accepted: 25 August 2017
First available in Project Euclid: 21 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1524276054

Digital Object Identifier
doi:10.1515/tmj-2017-0042

Mathematical Reviews number (MathSciNet)
MR3711857

Zentralblatt MATH identifier
1376.30012

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C50: Coefficient problems for univalent and multivalent functions

Keywords
Univalent functions bi-univalent functions bi-starlike function bi-convex function functions with bounded boundary rotation coefficient estimates convolution (Hadamard) product

Citation

Orhan, H.; Magesh, N.; Balaji, V. K. Certain classes of bi-univalent functions with bounded boundary variation. Tbilisi Math. J. 10 (2017), no. 4, 17--27. doi:10.1515/tmj-2017-0042. https://projecteuclid.org/euclid.tbilisi/1524276054


Export citation

References

  • R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344–351.
  • \c S. Alt\i nkaya and S. Yalçin, Second Hankel determinant for bi-starlike functions of order $\beta$, Matematiche (Catania) 71 (2016), no. 1, 115–125.
  • \c S. Alt\i nkaya and S. Yalçin, Coefficient problem for certain subclasses of bi-univalent functions defined by convolution, Math. Morav. 20 (2016), no. 2, 15–21.
  • S. Bulut, Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad J. Math. 43 (2013), no. 2, 59–65.
  • S. Bulut, N. Magesh and V. K. Balaji, Faber polynomial coefficient estimates for certain subclasses of meromorphic bi-univalent functions, C. R. Math. Acad. Sci. Paris 353 (2015), no. 2, 113–116.
  • M. Çağlar, E. Deniz, H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions, Turkish J. Math. 41 (2017), 694-706
  • M. Çağlar, H. Orhan and N. Yağmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27 (2013), no. 7, 1165–1171.
  • E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2 (2013), no. 1, 49–60.
  • P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften Series, 259, Springer Verlag, New York, 1983.
  • B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), no. 9, 1569–1573.
  • P. Goswami, B. S. Alkahtani and T. Bulboacă, Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions, arXiv:1503.04644v1 [math.CV] March (2015).
  • T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (2012), no. 4, 15–26.
  • J. M. Jahangiri, S. G. Hamidi and S. Abd. Halim, Coefficients of bi-univalent functions with positive real part derivatives, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 3, 633–640.
  • J. M. Jahangiri, N. Magesh and J. Yamini, Fekete-Szegö inequalities for classes of bi-starlike and bi-convex functions, Electron. J. Math. Anal. Appl., 3 (2015), no. 1, 133–140.
  • N. Magesh, T. Rosy and S. Varma, Coefficient estimate problem for a new subclass of biunivalent functions, J. Complex Anal. 2013, Art. ID 474231, 3 pp.
  • H. Orhan, N. Magesh and V. K. Balaji, Initial coefficient bounds for a general class of bi-univalent functions, Filomat, 29 (2015), no. 6, 1259–1267.
  • H. Orhan, N. Magesh and J. Yamini, Bounds for the second Hankel determinant of certain bi-univalent functions, Turkish J. Math. 40 (2016), no. 3, 679–687.
  • K. S. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975), 311–323.
  • S. Prema, B. S. Keerthi, Coefficient bounds for certain subclasses of analytic functions, Journal of Mathematical Analysis, 4 (2013), no. 1, 22–27.
  • C. Ramachandran, R. Ambrose Prabhu and N. Magesh, Initial coefficient estimates for certain subclasses of bi-univalent functions of Ma-Minda type, Applied Mathematical Sciences, 9 (2015), no. 47, 2299 – 2308.
  • H. M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc. 23 (2015), no. 2, 242–246.
  • H. M. Srivastava, S. Bulut, M. Çağlar, N. Ya ğmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27 (2013), no. 5, 831–842.
  • H. M. Srivastava, S. S. Eker and R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat 29 (2015), no. 8, 1839–1845.
  • H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some subclasses of $M$-fold symmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform. 41 (2015), 153–164.
  • H. M. Srivastava, S. Gaboury and F. Ghanim, Initial coefficient estimates for some subclasses of $m$-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed. 36 (2016), no. 3, 863–871.
  • H. M. Srivastava, S. B. Joshi, S. S. Joshi and H. Pawar, Coefficient estimates for certain subclasses of meromorphically bi-univalent functions, Palest. J. Math. 5 (2016), Special Issue: 1, 250–258.
  • H. M. Srivastava, N. Magesh and J. Yamini, Initial coefficient estimates for bi-$\lambda-$ convex and bi-$\mu-$ starlike functions connected with arithmetic and geometric means, Electronic J. Math. Anal. Appl. 2 (2014), no. 2, 152 – 162.
  • H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), no. 10, 1188–1192.
  • H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Global J. Math. Anal. 1 (2013), no. 2, 67–73.
  • H. M. Srivastava, S. Sivasubramanian and R. Sivakumar, Initial coefficient bounds for a subclass of $m$-fold symmetric bi-univalent functions, Tbilisi Math. J. 7 (2014), no. 2, 1–10.
  • H. Tang, G-T Deng and S-H Li, Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl. 317 (2013), 1–10.
  • H. Tang, H. M. Srivastava, S.Sivasubramanian and P. Gurusamy, The Fekete-Szegö functional problems for some subclasses of $m$-fold symmetric bi-univalent functions, J. Math. Inequal. 10 (2016), no. 4, 1063–1092.
  • A. E. Tudor, Bi-univalent functions connected with arithmetic and geometric means, J. Global Res. Math. Archives, 1 (2013), no. 3, 78–83.
  • Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), no. 6, 990–994.
  • Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), no. 23, 11461–11465.