Third-order differential inequalities and subordinations in the complex plane

There are over one hundred articles in the literature dealing with second-order differential inequalities and differential subordinations in the complex plane, and only two articles dealing with very narrow classes of third-order results. This article addresses a much more general class of third-order differential inequalities and subordinations. Let Ω and Δ be given sets in , let p be an analytic in the unit disc with . The authors determine the conditions on an admissible class of operators , so that the third-order differential subordination implies . Applications of these results to third-order differential inequalities, third-order differential subordinations, univalent functions and Bessel functions are presented.