On the classes of close-to-convex functions and close-to-star functions

Abstract

W.Kaplan〔1〕 introduced the class K of close-to-convex functions in the unit disk including its geometric characterization. Later, M.O. Reade〔7〕 introduced the class K* of close-to-star functions in the unit disk including its coefficient problem. In this paper, we obtain the following results by applying the method used by W.Kaplan〔1〕 and A.E. Livingston〔4〕; 1. f(z)∈K＊ if and only if for θ_(1)<θ_(2), 0≤r<1 ∫^(θ_(2))_(θ_(1))Re〔re^(iθ)ㆍf'(re^(iθ))/f(re^(iθ))〕dθ>-π. 2. If f(z) = z+□ a_(n)z^(n)∈K＊ then │a_(n)│≤n^(2) (n = 2, 3, ...) 3. If F is in K＊then f(z) =〔1/2〕〔zF(z)〕' is close-to-star for │z│<1/2. The result is sharp. ;본 논문은 W. Kalan이 소개한 close-to-convex 함수족과 M.0.Reade가 소개한 close-to-star 함수족을 A.E. Living ston의 논문과 결부시켜 연구함으로써 아래와 같은 세가지 결과를 얻게 되었다. 1. f(z)가 close-to-star 함수족에 속하게 되는 필요하고도 충분한 조건은 ∫^(θ2)_(θ1)Re[ re^(iθ)ㆍf'(re^(iθ))/f(re^(iθ))] dθ > -π 이다. 2. 만일 f(z) = z + □ a_(n)z^(n) ∈ K^(*) 이면 ｜a-(n)｜ ≤ n^(2) (n=2.3‥‥‥)이다. 3. 만일 F(z) ∈ K^(*) 이면 f (z) = [1/2][zF(z)]'은 ½인 반경을 갖는 경우에 close-to-star함수가 된다.