zbMATH — the first resource for mathematics

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.

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)