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Uniformly Convex Functions and a Corresponding Class of Starlike Functions

Proceedings of the American Mathematical Society (Impact Factor: 0.63). 05/1993; 118(1). DOI: 10.2307/2160026
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Available from: Frode Rønning, Dec 29, 2014
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    • "Moreover, this class for k = 1, corresponds to the class of uniformly convex functions UCV introduced by Goodman [5], which was studied extensively by Rønning [13], and independently also by Ma and Minda [12]. The class k-ST is related to the class k-U CV by means of the well-known Alexander equivalence between the usual classes of convex CV and starlike ST functions (see also the works [1] [8] [10] [7] [12] [13] concerning further developments involving each one of the classes k-U CV and k-ST ). The class k-ST has the geometric characterization (see [11]) that if f ∈ k-ST , then it maps a lens-like domain U (ζ, r) ∩ U (0, R) onto a starlike domain, where U (ζ, r) is a disk of radius r with center ζ , and 0 < R ≤ 1, |ζ | ≤ k, r ≥ |ζ | 2 + R 2 . "
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    • "Goodman proved the classical Alexander's result fðzÞ 2 UCV () zf 0 ðzÞ 2 UST , does not hold. On later, Rønning [7] introduced the class S p which consists of functions such that fðzÞ 2 UCV () zf 0 ðzÞ 2 S p . Also in [5], Rønning generalized the classes UCV and S p by introducing a parameter a in the following. "
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    • "Moreover for k = 1 it corresponds to the class k-ST was investigated in [4]. Note that the case k = 0 coincides with the usual case of starlike functions S * and if we take k = 1 we recover the class S p introduced by Rønning [15]. The properties of these classes have been studied among other in [3], [5], [6], [7], [17] and [20] "
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