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On uniformly convex functions. (English) Zbl 0744.30010
The author introduces the class $$UCV$$ of uniformly convex functions which consists of those convex functions $$f$$ transforming any circular arc in the unit disk $$E$$ with center $$\zeta\in E$$ into a convex arc. He shows that $$f\in UCV$$ if and only if $1+\hbox{Re}\left({f''(z) \over f'(z)}(z- \zeta)\right)\geq 0,$ for every pair $$(z,\zeta)\in E\times E$$, gives some examples of uniformly convex functions and proves some coefficient inequalities.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
circular arc; convex arc; uniformly convex functions
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