Second order differential inequalities in the complex plane

Let w(z) be regular in the unit disk U and let h(r, s, t) be a complex function defined in a domain of C³. The authors determine conditions on h such that $\text{¦ h(w(z), zw′(z), z}{}^2\text{w″(z))¦ < 1}$ implies $\text{¦ w(z)¦ < 1}$ and such that Re h(w(z), zw′(z), z²w″(z)) > 0 implies Re w(z) > 0. Applications of these results to univalent function theory, differential equations and harmonic functions are given.