Bulletin of the Iranian Mathematical Society

, Volume 44, Issue 1, pp 149–157 | Cite as

Faber Polynomial Coefficient Estimates for Bi-univalent Functions Defined by the Tremblay Fractional Derivative Operator

  • H. M. Srivastava
  • S. Sümer EkerEmail author
  • S. G. Hamidi
  • J. M. Jahangiri
Original Paper


Using the Tremblay fractional derivative operator in the complex domain, we introduce and investigate a new class of analytic and bi-univalent functions in the open unit disk. We use the Faber polynomial expansions to obtain upper bounds for the general coefficients of such functions subject to a gap series condition as well as obtaining bounds for their first two coefficients.


Tremblay fractional derivative operator Faber polynomials Analytic Univalent Bi-univalent functions 

Mathematics Subject Classification

Primary 30C45 Secondary 30C50 30C80 


  1. 1.
    Airault, H., Bouali, A.: Differential calculus on the Faber polynomials. Bull. Sci. Math. 130, 179–222 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Airault, H., Ren, J.: An algebra of differential operators and generating functions on the set of univalent functions. Bull. Sci. Math. 126, 343–367 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ali, R.M., Lee, S.K., Ravichandran, V., Supramaniam, S.: Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett. 25, 344–351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Çaglar, M., Orhan, H., Yagmur, N.: Coefficient bounds for new subclasses of bi-univalent functions. Filomat 27(7), 1165–1171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Deniz, E.: Certain subclasses of bi-univalent functions satisfying subordinate conditions. J. Class. Anal. 2(1), 49–60 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duren, P.L.: Univalent Functions Grundlehren Math. Wiss., vol. 259. Springer, New York (1983)Google Scholar
  7. 7.
    Eker, S.S.: Coefficient bounds for subclasses of m-fold symmetric bi-univalent functions. Turk. J. Math. 40, 641–646 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eker, S.S.: Coefficient estimates for new subclasses of m-fold symmetric bi-univalent functions. Theory Appl. Math. Comput. Sci. 6(2), 103–109 (2016)MathSciNetGoogle Scholar
  9. 9.
    Frasin, B.A., Aouf, M.K.: New subclasses of bi-univalent functions. Appl. Math. Lett. 24, 1569–1573 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamidi, S.G., Jahangiri, J.M.: Faber polynomial coefficient estimates for analytic bi-close-to-convex functions. C. R. Acad. Sci. Paris Ser. I(352), 17–20 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hamidi, S.G., Jahangiri, J.M.: Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations. Bull. Iran. Math. Soc. 41, 1103–1119 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamidi, S.G., Jahangiri, J.M.: Faber polynomial coefficient of bi-subordinate functions. C. R. Acad. Sci. Paris Ser. I(354), 365–370 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ibrahim, R.W., Jahangiri, J.M.: Boundary fractional differential equation in a complex domain. Bound. Value Probl. 2014(66), 11 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jahangiri, J.M., Hamidi, S.G.: Coefficient estimates for certain classes of bi-univalent functions. Int. J. Math. Math. Sci. 2013, Article ID 190560, 4 (2013)Google Scholar
  15. 15.
    Jahangiri, J.M., Hamidi, S.G.: Faber polynomial coefficient estimates for analytic bi-Bazilevic functions. Mat. Vesnik 67, 123–129 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Jahangiri, J.M., Hamidi, S.G., Halim, S.A.: Coefficients of bi-univalent functions with positive real part derivatives. Bull. Malays. Math. Sci. Soc. (2) 37, 633–640 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Jahangiri, J.M., Magesh, N., Yamini, J.: Fekete-Szego inequalities for classes of bi-starlike and biconvex functions. Electron. J. Math. Anal. Appl. 3, 133–140 (2015)MathSciNetzbMATHGoogle Scholar
  18. 17.
    Kumar, S.S., Kumar, V., Ravichandran, V.: Estimates for the initial coefficients of bi-univalent functions. Tamsui Oxf. J. Inf. Math. Sci. 29, 487–504 (2013)MathSciNetzbMATHGoogle Scholar
  19. 18.
    Magesh, N.N., Yamini, J.: Coefficient bounds for a certain subclass of bi-univalent functions. Int. Math. Forum 27, 1337–1344 (2013)CrossRefzbMATHGoogle Scholar
  20. 19.
    Srivastava, H.M.: Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions. In: Pardalos, P.M., Georgiev, P.G., Srivastava, H.M. (eds.) Nonlinear Analysis: Stability; Approximation; and Inequalities, Springer Optimization and Its Application 68, pp. 607–630. Springer, Berlin (2012)CrossRefGoogle Scholar
  21. 20.
    Srivastava, H.M., Bansal, D.: Coefficient estimates for a subclass of analytic and bi-univalent functions. J. Egypt. Math. Soc. 23, 242–246 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 21.
    Srivastava, H.M., Joshi, S.B., Joshi, S.S., Pawar, H.: Coefficient estimates for certain subclasses of meromorphically bi-univalent functions. Palest. J. Math. 5(Special Issue), 250–258 (2016)MathSciNetzbMATHGoogle Scholar
  23. 22.
    Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 23.
    Srivastava, H.M., Owa, S.: Univalent Functions, Fractional Calculus, and Their Applications. Ellis Horwood Ltd. Publ., Chichester (1989)zbMATHGoogle Scholar
  25. 24.
    Srivastava, H.M., Eker, S.S., Ali, R.M.: Coefficient Bounds for a certain class of analytic and bi-univalent functions. Filomat 29, 1839–1845 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 25.
    Todorov, P.G.: On the Faber polynomials of the univalent functions of class \(\sum \). J. Math. Anal. Appl. 162, 268–276 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 26.
    Tremblay, R.: Une Contribution ‘a la Théorie de la D érivée Fractionnaire, Ph.D. Thesis, Laval University, Québec (1974)Google Scholar
  28. 27.
    Xu, Q.H., Gui, Y.C., Srivastava, H.M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25, 990–994 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 28.
    Xu, Q.H., Xiao, H.G., Srivastava, H.M.: A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl. Math. Comput. 218, 11461–11465 (2012)MathSciNetzbMATHGoogle Scholar
  30. 29.
    Zaprawa, P.: On the Fekete–Szegö problem for classes of bi-univalent functions. Bull. Belg. Math. Soc. Simon Stevin 21(1), 169–178 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • H. M. Srivastava
    • 1
    • 2
  • S. Sümer Eker
    • 3
    Email author
  • S. G. Hamidi
    • 4
  • J. M. Jahangiri
    • 5
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of Medical Research, China Medical University HospitalChina Medical UniversityTaichungTaiwan, Republic of China
  3. 3.Department of Mathematics, Faculty of ScienceDicle UniversityDiyarbakirTurkey
  4. 4.Department of MathematicsBrigham Young UniversityProvoUSA
  5. 5.Department of Mathematical SciencesKent State UniversityBurtonUSA

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