Second-order differential superordination for analytic functions in the upper half-plane

Let \(\Omega\) be a set in the complex plane \(\mathbb{C}\). Also let \(p(z)\) be analytic in the upper halfplane \(\Delta=\{z:z\in\mathbb{C}\ \text{and}\ \Im(z) gt;0\}\) and suppose that \(\psi: \mathbb{C}^3\times\Delta\rightarrow\mathbb{C}\). In this paper, we investigate the problem of determining properties of functions \(p(z)\) that satisfy the following secondorder differential superordination: \[\Omega\subset\left\{\psi\left(p(z),p (z),p (z);z\right): z\in\Delta\right\}.\]Applications of these results to secondorder differential superordination for analytic functions in \(\Delta\) are also presented.