## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Coefficient bounds and convolution properties for certain classes of close-to-convex functions

#### Abstract

A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional $|a_3 - \mu a_2^2|$ for certain classes of univalent functions. In this paper, we consider the class $\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions which is defined by using subordination for analytic functions $\varphi$ and $\psi$ on the unit disc. Our main object is to provide bounds of the quantity $a_3 - \mu a_2^2$ for functions $f(z) = z + a_2 z^2 + a_3 z^3 + \dotsb$ in $\mathcal{C}(\varphi, \psi)$ in terms of $\varphi$ and $\psi$, where $\mu$ is a real constant. We also show that the class $\mathcal{C}(\varphi, \psi)$ is closed under the convolution operation by convex functions, or starlike functions of order $1/2$ when $\varphi$ and $\psi$ satisfy some mild conditions.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 76, Number 6 (2000), 95-98.

Dates
First available in Project Euclid: 23 May 2006

https://projecteuclid.org/euclid.pja/1148393497

Digital Object Identifier
doi:10.3792/pjaa.76.95

Mathematical Reviews number (MathSciNet)
MR1769977

Zentralblatt MATH identifier
0965.30006

#### Citation

Kim, Yong Chan; Choi, Jae Ho; Sugawa, Toshiyuki. Coefficient bounds and convolution properties for certain classes of close-to-convex functions. Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 6, 95--98. doi:10.3792/pjaa.76.95. https://projecteuclid.org/euclid.pja/1148393497

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