Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-06T12:01:29.194Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CLOSE-TO-CONVEX ANALOGUE OF CERTAIN STARLIKE FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  22 January 2020

VASUDEVARAO ALLU ,
JANUSZ SOKÓŁ  and
DEREK K. THOMAS
VASUDEVARAO ALLU*
Affiliation:
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha, India email avrao@iitbbs.ac.in
JANUSZ SOKÓŁ
Affiliation:
Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310Rzeszów, Poland email jsokol@ur.edu.pl
DEREK K. THOMAS
Affiliation:
Department of Mathematics, Swansea University, Singleton Park, SwanseaSA2 8PP, UK email d.k.thomas@swansea.ac.uk
*
✉avrao@iitbbs.ac.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.

For $f$ analytic in the unit disk $\mathbb{D}$, we consider the close-to-convex analogue of a class of starlike functions introduced by R. Singh [‘On a class of star-like functions’, Compos. Math.19(1) (1968), 78–82]. This class of functions is defined by $|zf^{\prime }(z)/g(z)-1|<1$ for $z\in \mathbb{D}$, where $g$ is starlike in $\mathbb{D}$. Coefficient and other results are obtained for this class of functions.

Keywords

univalent functionstarlike functionconvex functionalpha-convex functioncoefficient estimateintegral mean

MSC classification

Primary: 30C50: Coefficient problems for univalent and multivalent functions
Secondary: 30C55: General theory of univalent and multivalent functions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 268 - 281
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

References

Goodman, A. W., Univalent Functions, Vol. I (Mariner Publishing Co., Tampa, FL, 1983).Google Scholar
Goodman, A. W., Univalent Functions, Vol. II (Mariner Publishing Co., Tampa, FL, 1983).Google Scholar
Keogh, F. R. and Merkes, E. P., ‘A coefficient inequality for certain classes of analytic functions’, Proc. Amer. Math. Soc. 20(1) (1969), 8–12.CrossRefGoogle Scholar
Littlewood, J. E., Lectures on the Theory of Functions (Oxford University Press, Oxford, 1944).Google Scholar
Ma, W. C. and Minda, D., ‘A unified treatment of some special classes of unvalent functions’, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, Lecture Notes Anal. I (International Press, Cambridge, MA, 1994), 157–169.Google Scholar
Miller, S. S. and Mocanu, P. T., Differential Subordinations, Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, 225 (Marcel Dekker, New York, 2000).CrossRefGoogle Scholar
Pommerenke, Ch., Boundary Behaviour of Conformal Maps (Springer, Berlin, 1992).CrossRefGoogle Scholar
Privalov, I. I., ‘Sur les fonctions qui donnent la représentation conforme biunivoque’, Rec. Math. D. I. Soc. D. Moscou 31(3–4) (1924), 350–365.Google Scholar
Singh, R., ‘On a class of star-like functions’, Compos. Math. 19(1) (1968), 78–82.Google Scholar
Thomas, D. K., Tuneski, N. and Vasudevarao, A., Univalent Functions: A Primer, De Gruyter Studies in Mathematics, 69 (De Gruyter, Berlin, 2018).CrossRefGoogle Scholar