Subordination Results for Certain Subclasses of Uniformly Starlike and Convex Functions Defined by Convolution
In this paper we derive several subordination results for certain subclasses of uniformly starlike and convex functions defined by convolution.
On uniformly starlike functions
These are normalized functions regular and univalent in E: IzI < 1, for which f( E) is starlike with respect to the origin. Let y be a circle contained in E and let [ be the center of y. The Pinchuk question is this: Iff(z) is in ST, is it true thatf(y) is a closed curve that is starlike with respect tof(i)? In Section 5 we will see that the answer is no. There seems to be no reason to demand that the complete circle y lies in E, and we replace this condition with the stronger condition that y is an arc of a circle contained in E, but we still ask that [ the center of y is also in E. Thus we have,
UNIVERSAL LOCALLY COMPACT SCATTERED SPACES
If δ is an ordinal, we denote by C(δ) the class of all cardinal sequences of length δ of locally compact scattered (in short: LCS) spaces. If λ is an infinite cardinal, we write Cλ(δ) = {s ∈ C(δ) : s(0) = λ = min[s(ζ) : ζ < δ]}. An LCS space X is called Cλ(δ)-universal if SEQ(X) ∈ Cλ(δ), and for each sequence s ∈ Cλ(δ) there is an open subspace Y of X with SEQ(Y ) = s. We show that • there is a Cω(ω1)-universal LCS space, • under CH there is a Cω(δ)-universal LCS space for every ordinal δ < ω2, • under GCH for every infinite cardinal λ and every ordinal δ < ω2, there is a Cλ(δ)-universal LCS space, • there may exist a Cω(ω2)-universal LCS space. As a consequence, we obtain that it is consistent that 2 = ω2 and Cω(ω2) is large as possible, i.e. Cω(ω2) = {s ∈ 2{ω, ω1, ω2} : s(0) = ω}. 2000 Mathematics Subject Classification. 54A25, 06E05, 54G12, 03E35.