RETRACTED ARTICLE: Toeplitz Matrices Whose Elements are the Coefficients of Starlike and Close-to-Convex Functions



Let f be analytic in \(D=\{z: |z|< 1\}\) with \(f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}\). Suppose that \(S^*\) is the class of starlike functions, and K is the class of close-to-convex functions. The paper instigates a study of finding estimates for Toeplitz determinants whose elements are the coefficients \(a_{n}\) for f in \(S^*\) and K.


Univalent functions Coefficients Starlike Close-to-convex Toeplitz matrices 

Mathematics Subject Classification

30C45 30C50 



The authors wish to thank the referee for his/her comments and suggestions in improving the paper.


  1. 1.
    Duren, P. L.: Univalent Functions, (Springer-Verlag, 1983), 114–115. Mat. Sb.. 37 (79)(195) 471–476. (Russian) MR 17, 356Google Scholar
  2. 2.
    Hayman, W.K.: On the second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc. 3(18), 77–94 (1968)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Janteng, A., Halim, S., Darus, M.: Hankel determinants for starlike and convex functions. Int. J. Math. Anal. 1(13), 619–625 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Libera, R.J., Zlotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in P. Proc. Am. Math. Soc. 87(2), 251–257 (1983)MathSciNetMATHGoogle Scholar
  5. 5.
    Noonan, J.W., Thomas, D.K.: On the second Hankel determinant of a really mean p-valent functions. Trans. Am. Math. Soc. 223(2), 337–346 (1976)MATHGoogle Scholar
  6. 6.
    Pommerenke, Ch.: On the Hankel determinants of univalent functions. Mathematika (London) 16(13), 108–112 (1967)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ye, K., Lek-Heng, L.: Every matrix is a product of Toeplitz matrices. Found. Comput. Math. 14, 1–18 (2015)MATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2016

Authors and Affiliations

  1. 1.Department of MathematicsSwansea UniversitySwanseaUK
  2. 2.Institute of Mathematical SciencesUniversity of MalayaKuala LumpurMalaysia

Personalised recommendations