Coefficient bounds for bi-univalent classes defined by Bazilevič functions and convolution


In this paper, we obtain bi-univalent theorems for certain classes defined by convolution and Bazilevič functions with bounded boundary rotation. Also, we will find coefficients bounds for \(\ \left| a_{2}\right| ,\ \left| a_{3}\right| \ \) and \(\ \left| 2a_{2}^{2}h_{2}^{2}-a_{3}h_{3}\right| \ \) for the new classes \({\mathsf {W}} _{\alpha ,b,k,\delta }(f*h)\mathsf {\ }\)and \({\mathsf {M}}_{b,k,\delta }(f*h)\mathsf {.}\)

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  1. 1.

    Al-Oboudi, F.M., Haidan, M.M.: Spirallike function of complex order. J. Nat. Geometry 19, 53–72 (2000)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Altınkaya, S., Yalçın, S.: Coefficient problem for certain subclasses of bi-univalent functions defined by convolution. Math. Morav. 20(2), 15–21 (2016)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Aouf, M.K.: A generalization of functions with real part bounded in the mean on the unit disc. Math. Japn. 33(2), 175–182 (1988)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Aouf, M.K., Madian, S.M., Mostafa, A.O.: Bi-univalent properties for certain class of Bazilevič functions defined by convolution and with bounded boundary rotation. J. Egyptian Math. Soc. 27(11), 1–9 (2019)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bazilevič, I.E.: On a case of integrability in quadratures of the Lowner-Kufarev equation. Mat. Sb. 37(79), 471–476 (1955)

    MathSciNet  Google Scholar 

  6. 6.

    Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babeş-Bolyai Math. 31(2), 70–77 (1986)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Duren, P.L.: “Univalent Functions,” in Grundlehren der Mathematischen Wissenschaften Series, p. 259. Springer Verlag, New York (1983)

    Google Scholar 

  8. 8.

    El-Ashwah, R.M.: Subclasses of bi-univalent functions defined by convolution. J. Egypt. Math. Soc. 22, 348–351 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Frasin, B.A., Aouf, M.K.: New subclasses of bi-univalent functions. Appl. Math. Lett. 24(9), 1569–1573 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Goswami, P., Alkahtani, B. S., Bulboaca, T.: Estimate for initial Maclaurin coefficients of certain subclasses of bi-univalent functions, arXiv:1503.04644v1 [math.CV] March (2015)

  11. 11.

    Hayami, T., Owa, S.: Coefficient bounds for bi-univalent functions. PanAm. Math. J. 22(4), 15–26 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Motamednezhad, A., NosratiI, S., Zaker, S.: Bounds for initial Maclaurin coefficients of a subclass of bi-univalent functions associated with subordination. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1), 125–135 (2019)

    MathSciNet  Google Scholar 

  13. 13.

    Moulis, E.J.: Generalizations of the Robertson functions. Pac. J. Math. 81(1), 167–174 (1979)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Nasr, M.A., Aouf, M.K.: Starlike functions of complex order. J. Nat. Sci. Math. 25, 1–12 (1985)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Nasr, M.A., Aouf, M.K.: Functions of bounded boundary rotation of complex order. Rev. Roum. Math. Pure Appl. 32(7), 623–629 (1987)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Noor, K., Arif, M., Muhammad, A.: Mapping properties of some classes of analytic functions under an integral operator. J. Math. Inequal. 4(4), 593–600 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Orhan, H., Magesh, N., Balaji, V.K.: Certain classes of bi-univalent functions with bounded boundary variation. Tbili. Math. J. 4(10), 17–27 (2017)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Padmanabh, K.S., Paravatham, R.: Properties of a class of functions with bounded boundary rotation. Ann. Polon. Math. 31(3), 842–853 (1975)

    MathSciNet  Google Scholar 

  19. 19.

    Pinchuk, B.: Functions of bounded boundary rotation. Isr. J. Math. 10, 7–16 (1971)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Prema, S., Keerthi, B.S.: Coefficient bounds for certain subclasses of analytic functions. J. Math. Anal. 4(1), 22–27 (2013)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Robertson, M.S.: Variational formulas for several classes of analytic functions. Math. Z. 118, 311–319 (1976)

    Google Scholar 

  22. 22.

    Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Taha, T. S.: Topics in univalent function theory, Ph. D. Thesis, University of London, (1981)

  24. 24.

    Umarani, P.G., Aouf, M.K.: Linear combination of functions of bounded boundary rotation of order \(\alpha\). Tamkang J. Math. 20(1), 83–86 (1989)

    MathSciNet  MATH  Google Scholar 

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Aouf, M.K., Madian, S.M. Coefficient bounds for bi-univalent classes defined by Bazilevič functions and convolution. Bol. Soc. Mat. Mex. 26, 1045–1062 (2020).

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  • Bi-univalent
  • Bazilevič functions
  • Hadamard product
  • Bounded boundary rotation

Mathematics Subject Classification

  • 30C45