Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane

Abstract

There are many articles in the literature dealing with a first, second and third-order differential inequalities, inclusions or subordinations in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential inequalities, inclusions and subordinations in two complex functions p and q in the complex plane. A typical example of a system of differential inequalities is

$$ \left\{ {\begin{array}{*{20}l} {\text{Re} [2p(z) + zp^{{\prime }} (z) + z^{2} p^{{\prime \prime }} (z) - q(z)] > 0,} \hfill \\ {\text{Re} [p(z) + 7zq^{{\prime }} (z)] > 4.} \hfill \\ \end{array} } \right. $$

The authors determine properties of the functions p and q satisfying some special systems of differential inequalities, and extend their results to differential inclusions and subordinations.