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Some simple criteria for starlikeness and convexity. (English) Zbl 0793.30008
Let $$A_ n$$ denote the class of functions $$f(z)= z+ a_{n+1} z^{n+1}+\cdots$$, $$n\geq 1$$, 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 $$U$$. $$I_ c: A_ n\to A_ n$$ is the integral operator defined by $F(z)= I_ c(f)(z)= (c+ 1)\int^ 1_ 0 f(tz)t^{c-1} dt,\quad z\in U.$ Note that $$F\in K\Leftrightarrow f\in S^*$$ 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 $$f\in A_ n$$ or of $$F= I_ c(f)$$. For $$f\in A_ n$$, $$c>-1$$, the conditions imposed to $$f'$$ or $$f''$$ are of the form $$| f'(z)- 1|<M$$, $$| f''(z)|< M$$ or $$|\arg f'(z)|< M$$, 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:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
starlikeness criteria; subordinations; convexity criteria