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Some simple criteria for starlikeness and convexity. (English) Zbl 0793.30008
Let $A\sb n$ denote the class of functions $f(z)= z+ a\sb{n+1} z\sp{n+1}+\cdots$, $n\ge 1$, that are analytic in the unit disk $U$ and let $S\sp*$ be the class of starlike functions and $K$ the class of convex functions in $U$. $I\sb c: A\sb n\to A\sb n$ is the integral operator defined by $$F(z)= I\sb c(f)(z)= (c+ 1)\int\sp 1\sb 0 f(tz)t\sp{c-1} dt,\quad z\in U.$$ Note that $F\in K\Leftrightarrow f\in S\sp*$ 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 $f\in A\sb n$ or of $F= I\sb c(f)$. For $f\in A\sb n$, $c>-1$, the conditions imposed to $f'$ or $f''$ are of the form $\vert f'(z)- 1\vert<M$, $\vert f''(z)\vert< M$ or $\vert\arg f'(z)\vert< M$, where $M$ is a suitable constant, depending on $n$ (and $c$ for $I\sb c(f)$). Some interesting examples, which point out the usefulness of the new starlikeness and convexity criteria, are given.

30C45Special classes of univalent and multivalent functions