## 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}- μa_{2}^{2}| for certain classes of univalent functions. In this paper, we consider the class C(φ, ψ) of normalized close-to-convex functions which is defined by using subordination for analytic functions φ and ψ on the unit disk. Our main object is to provide bounds of the quantity a_{3} - μa_{2}^{2} for functions f(z) = z + a_{2}Z^{2} + a_{3}z^{3} + ⋯ in C(φ, ψ) in terms of φ and ψ, where μ is a real constant. We also show that the class C(φ, ψ) is closed under the convolution operation by convex functions, or starlike functions of order 1/2 when φ and ψ satisfy some mild conditions.

Original language | English |
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Pages (from-to) | 95-98 |

Number of pages | 4 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 76 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2000 |

## Keywords

- Coefficient bound
- Convolution
- Univalent function