Vol. 44, No. 1, 1973

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On close-to-convex functions of order β

James Waring Noonan

Vol. 44 (1973), No. 1, 263–280

For β 0, denote by K(β) the class of normalized functions f, regular and locally schlicht in the unit disc, which satisfy the condition that for each r < 1, the tangent to the curve C(r) = {f(rei𝜃) : 0 𝜃 < 2π} never turns back on itself as much as βπ radians. K(β) is called the class of close-to-convex functions of order β. The purpose of this paper is to investigate the asymptotic behavior of the integral means and Taylor coefficients of f K(β). It is shown that the function Fβ, given by Fβ(z) = (1(2(β + 1))){((1 + z)(1 z))β+1 1}, is in some sense extremal for each of these problems. In addition, the class B(α) of Bazilevic functions of type α > 0 is related to the class K(1∕α). This leads to a simple geometric interpretation of the class B(α) as well as a geometric proof that B(α) contains only schlicht functions.

Mathematical Subject Classification
Primary: 30A32
Received: 30 June 1971
Published: 1 January 1973
James Waring Noonan