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Classes of Uniformly Convex and Uniformly Starlike Functions as Dual Sets

✍ Scribed by I.R. Nezhmetdinov

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 170 KB
Volume: 216
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.1997.5640

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✦ Synopsis

In this paper the classes of uniformly convex and uniformly starlike functions are Ž presented as dual sets for certain function families in the sense of convolution . theory . The results are used to find some sharp sufficient conditions for functions, regular in the unit disk, to belong to the above classes.

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Let z be an analytic function with positive real part on ⌬ s z; z -1 with Ž . Ž . 0 s 1, Ј 0 ) 0 which maps the unit disk ⌬ onto a region starlike with respect Ž . to 1 and symmetric with respect to the real axis. Let ST denote the class of Ž . Ž . Ž . Ž . Ž . Ž . analytic functions f z with f 0 s