Remarks on a Multiplier Conjecture for Univalent Functions
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چکیده
In this paper we study some aspects of a conjecture on the convolution of univalent functions in the unit disk D , which was recently proposed by Grünberg, Ronning, and Ruscheweyh (Trans. Amer. Math. Soc. 322 (1990), 377-393) and is as follows: let 3 := {/ analytic in D: \f"(z)\ < Ref'(z), z £ D} and g, h £ S? (the class of normalized univalent functions in D) . Then Ke(f*g*h)(z)/z > 0 in D . We discuss several special cases, which lead to interesting, more specific statements about functions in S* , determine certain extreme points of 3 , and note that the former conjectures of Bieberbach and Sheil-Small are contained in this one. It is an interesting matter of fact that the functions in 3 , which are "responsible" for the Bieberbach coefficient estimates are not extreme points in 3 .
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تاریخ انتشار 2010