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

ABSTRACT A function f(z)=z+a 2 z 2 +⋯ is said to be uniformly convex in the unit disk U if (i) it is starlike in U, and (ii) for each circular arc γ in U with centre ζ in U, the arc f(γ) is convex; this class of functions was introduced by A. W. Goodman [J. Math. Anal. Appl. 155, No. 2, 364-370 (1991; Zbl 0726.30013)]. The author then defines a related class S p ={F:F starlike in U, F(z)=zf ' (z), f uniformly convex in U}. The author proves that a function f(z)=z+a 2 z 2 +⋯, analytic in U, belongs to S p if and only if |zf ' (z)/f(z)-1|<Rezf ' (z)/f(z) for all z∈U, looks at particular examples of functions in S p such as f(z)=z+z n /(2n-1), and proves that, if f is a normalised univalent function, then 1 rf(rz) belongs to S p if and only if r≤0·33217⋯. Finally, he proves that, if f(z)=z+a 2 z 2 +⋯ belongs to S p , then |a 2 |≤8/π 2 and |a n |≤8 (n-1)π 2 ∏ k=3 n (1+8 (k-2)π 2 ) for n≥3; also, that |f(z)/z|≤exp(14ζ(3)/π 2 ) (where ζ is the Riemann Zeta function), and that this inequality is best-possible.

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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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