Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-02T21:07:18.708Z Has data issue: false hasContentIssue false
  • English
  • Français

TOEPLITZ DETERMINANTS WHOSE ELEMENTS ARE THE COEFFICIENTS OF ANALYTIC AND UNIVALENT FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  26 February 2018

MD FIROZ ALI ,
D. K. THOMAS  and
A. VASUDEVARAO
MD FIROZ ALI
Affiliation:
Indian Statistical Institute, Chennai Centre, 37, Nelson Manickam Road, Aminjikarai, Chennai-600 029, Tamilnadu, India email ali.firoz89@gmail.com
D. K. THOMAS
Affiliation:
Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK email d.k.thomas@swansea.ac.uk
A. VASUDEVARAO*
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India email alluvasu@maths.iitkgp.ernet.in
*
✉alluvasu@maths.iitkgp.ernet.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.

Keywords

analytic functionunivalent functionstarlike functionconvex functionclose-to-convex functiontypically real functionToeplitz determinant

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 2 , April 2018 , pp. 253 - 264
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Aleman, A. and Constantin, A., ‘Harmonic maps and ideal fluid flows’, Arch. Ration. Mech. Anal. 204 (2012), 479–513.Google Scholar
Constantin, O. and Martin, M. J., ‘A harmonic maps approach to fluid flows’, Math. Ann. 316 (2017), 1–16.Google Scholar
Duren, P. L., Univalent Functions, Grundlehren der mathematischen Wissenschaften, 259 (Springer, New York–Berlin–Heidelberg–Tokyo, 1983).Google Scholar
Efraimidis, I., ‘A generalization of Livingston’s coefficient inequalities for functions with positive real part’, J. Math. Anal. Appl. 435(1) (2016), 369–379.Google Scholar
Janteng, A., Halim, S. A. and Darus, M., ‘Hankel determinant for starlike and convex functions’, Int. J. Math. Anal. 1 (2007), 619–625.Google Scholar
Keogh, F. R. and Merkes, E. P., ‘A coefficient inequality for certain classes of analytic functions’, Proc. Amer. Math. Soc. 20 (1969), 8–12.Google Scholar
Ma, W., ‘Generalized Zalcman conjecture for starlike and typically real functions’, J. Math. Anal. Appl. 234(1) (1999), 328–339.CrossRefGoogle Scholar
Radhika, V., Sivasubramanian, S., Murugusundaramoorthy, G. and Jahangiri, J. M., ‘Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation’, J. Complex Anal. 2016 (2016), 4960704.Google Scholar
Robertson, M. S., ‘On the coefficients of a typically-real function’, Bull. Amer. Math. Soc. 41 (1935), 565–572.CrossRefGoogle Scholar
Szapiel, W., Extremal Problems for Convex Sets. Applications to Holomorphic Functions (in Polish), Dissertation Ann. Univ. Mariae Curie-Skłodowska, Sect. A. vol. 37 (UMCS Press, Lublin, 1986).Google Scholar
Thomas, D. K. and Halim, S. A., ‘Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions’, Bull. Malays. Math. Sci. Soc. 40(4) (2017), 1781–1790.Google Scholar
Ye, K. and Lim, L.-H., ‘Every matrix is a product of Toeplitz matrices’, Found. Comput. Math. 16 (2016), 577–598.Google Scholar