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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.)
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