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Some simple criteria for starlikeness and convexity. (English) Zbl 0793.30008

Let A n denote the class of functions f(z)=z+a n+1 z n+1 +, n1, that are analytic in the unit disk U and let S * be the class of starlike functions and K the class of convex functions in U. I c :A n A n is the integral operator defined by

F(z)=I c (f)(z)=(c+1) 0 1 f(tz)t c-1 dt,zU·

Note that FKfS * if c=0. Using the theory of differential subordinations, the author proves some criteria involving f ' or f '' only, in determining the starlikeness or convexity of fA n or of F=I c (f). For fA n , c>-1, the conditions imposed to f ' or f '' are of the form |f ' (z)-1|<M, |f '' (z)|<M or |argf ' (z)|<M, where M is a suitable constant, depending on n (and c for I c (f)). Some interesting examples, which point out the usefulness of the new starlikeness and convexity criteria, are given.

MSC:
30C45Special classes of univalent and multivalent functions