Classes of univalent integral operators (Q805790)

If \(f(z)=z+a_ 2z^ 2+..\). is analytic in the disk U, then \[ I(f)(z)=[\frac{\beta +\gamma}{z^{\gamma}\Phi (z)}\int^{z}_{0}f^{\alpha}(t)\phi (t)t^{\delta - 1}dt]^{1/\beta}=z+b_ 2z^ 2+... \] is univalent (and starlike or convex) for suitable choices of the analytic functions \(\phi\) (z) and \(\Phi\) (z) and parameters \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). In various forms this operator has been studied by many investigators as is well-documented in the authors' presentation. The purpose of the paper is to allow the parameters \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) to be complex and to admit greater variety in the choices of f(z). The authors make clever use of subordination techniques developed here which follow from their earlier papers on integral operators.