Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T02:35:08.159Z Has data issue: false hasContentIssue false
  • English
  • Français

ON OZAKI CLOSE-TO-CONVEX FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  20 September 2018

VASUDEVARAO ALLU Open the ORCID record for VASUDEVARAO ALLU [Opens in a new window] ,
DEREK K. THOMAS Open the ORCID record for DEREK K. THOMAS [Opens in a new window]  and
NIKOLA TUNESKI Open the ORCID record for NIKOLA TUNESKI [Opens in a new window]
VASUDEVARAO ALLU*
Affiliation:
NFA-18, IIT Campus, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India email [email protected]
DEREK K. THOMAS
Affiliation:
Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK email [email protected]
NIKOLA TUNESKI
Affiliation:
Faculty of Mechanical Engineering, Ss. Cyril and Methodius University in Skopje, Karpos 2 bb, 1000 Skopje, Republic of Macedonia email [email protected]
*
✉[email protected]
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 $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

Keywords

analyticunivalentstrongly close-to-convex functionscoefficient estimates

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 99 , Issue 1 , February 2019 , pp. 89 - 100
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Ali, R. M., ‘Coefficients of the inverse of strongly starlike functions’, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), 63–71.Google Scholar
Ali, R. M. and Singh, V., ‘On the fourth and fifth coefficients of strongly starlike functions’, Results Math. 29(3–4) (1996), 197–202.Google Scholar
Brannan, D. A., Clunie, J. and Kirwan, W. E., ‘Coefficient estimates for a class of star-like functions’, Canad. J. Math. 22 (1970), 476–485.Google Scholar
Goodman, A. W., Univalent Functions, Vol. I (Mariner, Tampa, FL, 1983).Google Scholar
Kaplan, W., ‘Close-to-convex schlicht functions’, Michigan Math. J. 1 (1952), 169–186.Google Scholar
Kargar, R. and Ebadian, A., ‘Ozaki’s conditions for general integral operator’, Sahand Commun. Math. Anal. (SCMA) 5(1) (2017), 61–67.Google Scholar
Keogh, F. R. and Miller, S. S., ‘On the coefficients of Bazilevič functions’, Proc. Amer. Math. Soc. 30 (1971), 492–496.Google Scholar
Miller, S. S. and Mocanu, P. T., Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 225 (Marcel Dekker, New York, 2000).Google Scholar
Ozaki, S., ‘On the theory of multivalent functions’, Sci. Rep. Tokyo Bunrika Daigaku 4 (1941), 455–486.Google Scholar
Pommerenke, C., ‘On starlike and close-to-convex functions’, Proc. Lond. Math. Soc. (3) 13(3) (1963), 290–304.Google Scholar
Ponnusamy, S., Sahoo, S. K. and Yanagihari, H., ‘Radius of convexity of partial sums in the close-to-convex family’, Nonlinear Anal. 95 (2014), 219–228.Google Scholar
Privalov, I. I., ‘Sur les fonctions qui donnent la représentation conforme biunivoque’, Rec. Math. Soc. Moscou 31 (1924), 350–365; Russian translation Mat. Sb. 31(3–4) (1924), 350–365.Google Scholar
Strohhäcker, E., ‘Beiträge zur Theorie der schlichten Funktionen’, Math. Z. 37(1) (1933), 356–380.Google Scholar
Suffridge, T. J., ‘Some special classes of conformal mappings’, in: Handbook of Complex Analysis: Geometric Function Theory, Vol. 2 (ed. Kühnau, R.) (Elsevier, Amsterdam, 2005), 309–338.Google Scholar
Thomas, D. K., ‘On starlike and close-to-convex univalent functions’, J. Lond. Math. Soc. (2) 42 (1967), 427–435.Google Scholar
Thomas, D. K., ‘A note on starlike functions’, J. Lond. Math. Soc. (2) 43 (1968), 703–706.Google Scholar
Thomas, D. K. and Verma, S. S., ‘Invariance of the coefficients of strongly convex functions’, Bull. Aust. Math. Soc. 95 (2017), 436–445.Google Scholar